LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


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PHYSICAL    MEASUREMENTS 


IN 


PROPERTIES   OF   MATTER 
AND   HEAT 


BY 

ELMER    E.  HALL 

Assistant  Professor  of  Physics,  University  of  California 
AND 

T.  SIDNEY  ELSTON 

Instructor  in  Physics,  University  of  California 


BERKELEY,   CALIFORNIA 
1908 


Entered  According  to  Act  of  Congress  in  the  Year  1908,  by 

ELMER  E.  HALL  and  T.  SIDNEY  ELSTON, 
In  the  Office  of  the  Librarian  of  Congress  at  Washington. 


Printed  by  The  Times  Publishing  Company 
Palo  Alto,  California 


PREFACE. 

This  manual  is  printed  for  the  use  of  students  in  the  Fresh- 
man year  in  the  various  colleges  in  the  University  of  Califor- 
nia, and  represents  the  laboratory  side  of  a  three-unit  course 
consisting  of  one  lecture,  one  recitation,  and  one  laboratory 
period  per  week  throughout  the  year.  The  course  is  preceded 
by  a  matriculation  course  in  Elementary  Physics  and  is  a  part 
of  a  two-years  course  in  General  Physics,  the  work  in  Sound, 
Light,  and  Electricity  being  given  during  the  Sophomore 
year.  The  present  form  of  the  course  is  a  revision  of  the  work 
as  given  heretofore.  The  predecessors  of  the  present  writers, 
Professor  Harold  Whiting,  A.  C.  Alexander,  G.  K.  P>urgess, 
Bruce  V.  Hill,  and  A.  S.  King,  have  all  left  their  imprint  on 
the  course.  The  writers  are  especially  indebted  to  Dr.  King, 
whose  mimeographed  directions  they  have  freely  used.  Sev- 
eral experiments  have  been  taken  from  the  Sophomore  course 
of  Professor  E.  R.  Drew,  written  before  the  present  division 
of  subjects  was  made.  Free  use  has  been  made  of  the  pub- 
lished texts  and  manuals,  as  is  indicated  by  the  references 
given. 

ELMSR  E.   HALL, 

Berkeley,  Cal.,  July,   1908.  T.   SIDNEY  ELSTON. 


205947 


LIST    OF    EXPERIMENTS. 

1.  Use  of  the  Balance.     Density  of  a  Solid. 

2.  Density  by  Jolly's  Balance.     Archimedes'  Principle. 

3.  Boyle's  Law. 

4.  The  Volumenometer. 

5.  Density  of  Air. 

6.  Relative  Density  of  Carbon  Dioxide. 

7.  The  Force  Table. 

8.  Equilibrium  Conditions  for  Three  Forces  at  a  Point. 

9.  Uniformly  Accelerated  Motion. 

10.  The  Force  Equation. 

11.  The  Simple  Pendulum. 

12.  The  Principle  of  Moments. 

13.  The  Model  Balance. 

14.  Surface  Tension  by  Direct  Measurement. 

15.  Capillarity.     Rise  of  Liquids  in  Tubes. 

1 6.  Rise  of  Liquids  Between  Plates. 

17.  Pressure  and  Radius  of  a  Soap-Bubble. 

1 8.  Viscosity.     Flow  of  Liquids  in  Tubes. 

19.  Relative  Densities  of  Gases.     Time  of  Efflux. 

20.  Calibration  of  a  Thermometer,  Absolute. 

21.  Calibration  of  a  Thermometer,  Relative. 

22.  Variation  of  Boiling  Point  with  Pressure. 

23.  Expansion  of  a  Liquid  by  Archimedes'  Principle. 

24.  Comparison  of  Alcohol  and  Water  Thermometers. 

25.  Expansion  of  Mercury  by  Regnault's  Method. 

26.  Expansion  of  Glass  by  Weight  Thermometer. 

27.  Expansion  of  a  Liquid  by  Pycnometer  Method. 

28.  Expansion  Curve  of  Water. 

29.  Specific  Heat  of  a  Liquid  by  Method  of  Heating. 

30.  Specific  Heat  of  a  Liquid  by  Method  of  Cooling. 


LIST  OF  EXPERIMENTS  V 

.31.  Mechanical   Equivalent  of  Heat  by   Method  of  Percus- 
sion. 

32.  Mechanical  Equivalent  of  Heat  by  Method  of  Puluj. 

33.  Cooling  through  Change  of  State. 

34.  Heat  of  Fusion. 

35.  Heat  of  Vaporization  at  Boiling  Point. 

36.  Heat  of  Vaporization  at  Room  Temperature. 

37.  Freezing  Point  of  Solutions. 

38.  Heat  of  Solution. 

39.  Heat  of  Chemical  Combination. 

40.  Expansion  of  Air  at  Constant  Pressure  by  Flask  Method. 

41.  Expansion  of  Air  at  Constant  Pressure.     Constant-Pres- 

sure Air-Thermometer. 

42.  Constant- Volume  Air-Thermometer. 

43.  Variation  of  Pressure,  Volume,  and  Temperature  of  a 

Saturated  Vapor. 

44.  Hygrometry. 

45.  Density  of  the  Air  by  the  Barodeik. 

46.  Coefficient  of  Friction. 

47.  Conservation  of  Momentum.     Coefficient  of  Restitution. 

48.  Young's  Modulus  by  Stretching. 

49.  Hooke's  Law  for  Twisting.     Coefficient  of  Rigidity. 

50.  Centripetal  Force. 

51.  Friction  Brake.     Power  Supplied  by  a  Motor. 

52.  Absorption  and  Radiation. 

53.  Ratio  of  the  Two  Principal  Specific  Heats  of  a  Gas. 


REFERENCES. 

References  are  given  throughout  the  manual  to  the  follow- 
ing books.  Except  in  cases  of  confusion  the  reference  gives 
the  author's  name,  omitting  the  title  of  the  book. 

Ames  and  Bliss,  A  Manual  of  Experimental  Physics. 

Edser,  Heat  for  Advanced  Students. 

Ferry  and  Jones,  A  Manual  of  Practical  Physics,  Vol.  I. 

Franklin,  Crawford,  and  MacNutt,  Practical  Physics, 
Vol.  I. 

Glazebrook  and  Shaw,  Practical  Physics  (Third  Edition). 

Hastings  and  Beach,  General  Physics. 

Kohlrausch,  Physical  Measurements  (Third  English  Edi- 
tion). 

Miller,  Laboratory  Physics. 

Millikan,  Molecular  Physics  and  Heat. 

Preston,  Theory  of  Heat  (Second  Edition). 

Watson,  Text-Book  of  Physics   (Fourth  Edition,  1903). 

Watson,  Text-Book  of  Practical  Physics. 


PHYSICAL    MEASUREMENTS, 


PROPERTIES  OF  MATTER  AND  HEAT. 

This  book  is  intended  to  be  mainly  a  manual  of  directions. 
It  is  expected  that  the  student  will  consult  the  manuals  given 
in  the  list  of  references  (a  liberal  number  of  copies  of  which 
are  available)  for  general  notions  regarding  physical  measure- 
ments, the  discussion  of  results,  the  effect  of  errors  in  obser- 
vation and  methods  for  their  complete  or  partial  elimination. 

Mimeographed  directions  regarding  the  method  of  writing 
up  and  handing  in  the  record  of  the  experiments,  a  list  con- 
taining the  required  experiments,  and  the  order  in  which  they 
are  to  be  performed  will  be  given  each  student.  The  satis- 
factory completion  of  thirty  experiments  fulfills  the  labora- 
tory part  of  the  requirement  for  the  year's  work. 


i.     USE  OF  THE  BALANCE.    DENSITY  OF  A  SOLID. 

References. — Glazebrook    and    Shaw,    p.    91;    Miller,    p.    55;    Kohl- 
rausch,  p.  30;  Franklin,  Crawford,  and  MacNutt,  p.  23. 

In  weighing  with  a  sensitive  balance  the  indications  are 
made  by  means  of  a  long  pointer  attached  to  the  beam  and 
arranged  to  vibrate  in  front  of  a  fixed  scale.  If  the  balance 
is  sensitive  the  pointer  will  swing  many  times  back  and  forth 
before  it  finally  comes  to  rest  at  a  definite  point  which  marks 
the  position  of  equilibrium.  Time  would  be  wasted  in  waiting 
for  it  to  stop,  and  even  then  the  indications  of  the  moving 
pointer  are  more  trustworthy  than  those  of  one  which  has 
come  to  rest,  because  the  latter  may  not  be  in  its  true  position 


2  USE  OF  THE  BALANCE.      DENSITY  OF  A  SOLID.  f  I 

of  equilibrium,  or  rest-point,  owing  to  friction.  To  obtain 
the  rest-point,  the  pointer  is  allowed  to  vibrate  and  the  turning- 
points  of  a  number  of  consecutive  swings  are  read,  the  num- 
ber being  so  chosen  as  to  give  an  even  number  of  turning 
points  on  one  side  and  an  odd  number  on  the  other.  A  little 
consideration  will  show  that  under  this  condition  the  point 
halfway  between  the  mean  of  all  the  left-hand  and  the  mean 
of  all  the  right-hand  readings  is  the  true  rest-point.  This  way 
of  getting  the  rest-point  is  known  as  the  "method  of  vibra- 
tions.'' 

The  sensitive  balance  must  be  handled  with  the  greatest 
care,  since  any  jarring  or  rapid  vibration  of  the  beam  is  likely 
to  injure  the  knife-edges  upon  which  the  beam  rests.  On  this 
account  the  beam  should  be  lowered  each  time  before  a  mass 
is  placed  on  the  pan  or  removed  from  it,  and  also  when  the 
weighing  is  completed. 

To  illustrate  the  use  of  the  balance,  let  it  be  required  to 
find  the  density  of  a  solid  (a  hard  rubber  cylinder,  for  ex- 
ample). 

(a)  With  the  pans  of  the  balance  empty,   raise  the  beam 
slowly  and  allow  the  pointer  to  swing  over  four  or  five  scale 
divisions.    Take  and  record  an  even  number  of  turning  points 
on  one  side  and  an  odd  number  on  the  other  (respectively  4 
and  3,  say),  and  from  them  determine  the  rest-point.     Make 
two  determinations  in  this  way,  and  take  the  mean  as  the  zero 
rest-point.    The  door  to  the  glass  case  should  always  be  closed 
when  determining  the  rest-point. 

(b)  Place  the  hard  rubber  cylinder  on  the  left-hand  pan  of 
the  balance,  and  add  masses  to  the  other  pan  until  the  pointer 
does  not  swing  off  the  scale  when  the  beam  is  raised.     In 
making  trials   for  the  correct   mass   on   the   right-hand   pan, 
raise  the  beam  only  high  enough  to  see  which  side  has  the 
greater  mass,  in  order  to  avoid  violent  rocking  of  the  beam. 
Use  the  fractional  masses  to  bring  the  pointer  approximately 
to  the  zero  rest-point,  and  then  determine  the  rest-point  by 


i]  USE;  OF  THE  BALANCE:.    DE)NSITY  OF  A  SOLID.  3 

the  method  of  vibrations.  To  determine  how  much  must  be 
added  to,  or  subtracted  from,  the  masses  on  the  right-hand 
pan  to  take  account  of  the  fact  that  the  rest-point  with  the 
loaded  balance  does  not  coincide  with  the  zero  rest-point,  add 
a  5  or  10  mg.  mass  to  either  pan  and  determine  the  sensitive- 
ness of  the  balance, — that  is,  determine  the  mass  which,  added 
to  either  pan,  will  move  the  pointer  through  one  division. 
From  this,  the  difference  in  rest-points,  and  the  masses  in  the 
right-hand  pan,  find  the  exact  mass  which  will  balance  the 
hard  rubber  cylinder  in  air.  (In  fine  weighing  it  is  often  con- 
venient to  use  a  centigram  "rider."  By  properly  placing  the 
rider  on  the  graduated  scale  attached  to  the  beam,  the  equiv- 
alent of  any  desired  mass  from  i  to  10  mg.  may  be  added  to 
either  pan.  Final  adjustments  can  thus  be  made  without  open- 
ing the  balance  case.  The  rider  should  never  be  moved 
without  first  lowering  the  balance  beam,.) 

(c)  Unless  the  body  whose  mass  is  sought  has  the  same 
specific  weight  as  the  masses  used  to  balance  it,  the  body  will 
be  buoyed  up  by  the  air  either  more  or  less  than  the  masses 
are  buoyed  up,  and  this  will  introduce  an  error  which  is  by 
no  means  negligible  in  careful  measurements.     To  correct  for 
air-buoyancy:     Measure  the  dimensions  of  the  cylinder  with 
vernier  calipers,  and  compute  its  volume.     Calculate  the  vol- 
ume of  the  standard  masses  from  their  marked  values  and  the 
density  of  brass  (8.4  gms.  per  cc.).     From  these  volumes  and 
the  density  of  air  at  the  temperature  and  barometric  pressure 
at   the  time  of  the   experiment    (see   Tables),   determine   the 
correction  for  air-buoyancy.     Calculate  the  density  of  the  cyl- 
inder. 

(d)  If  the  arms  of  the  balance  are     unequal     in     length, 
"double  weighing"   is   necessary.      Place  the  cylinder   in   the 
right-hand  pan  and  find  the  mass  as  before.     The  true  mass 
is  then  given  by  m=\/ni1m2,  where  mt  and  m2  are  the  values 
obtained  by  the  two  weighings.    The  proof  of  this  by  an  appli- 
cation of  the  principle  of  moments  is  left  to  the  student. 


4  DENSITY   OF  A   SOLID   BY   JOLLY'S    P.. \I..\\CK.  [2 

2.    DENSITY  OF  A  SOLID  BY  JOLLY'S  BALANCE. 

References. — Ames  and  Bliss,  p.  197;  Ferry  and  Jones,  p.  97. 

Jolly's  balance  consists  of  a  long"  spiral  spring  suspended  in 
front  of  a  graduated  mirror.  As  adapted  to  this  experiment 
the  upper  end  of  the  spring  may  be  raised  or  lowered :  the 
lower  end  carries  two  light  pans.  The  lower  of  the  two  pan- 
is  always  immersed  in  a  beaker  of  water  standing  on  a  small 
platform  which  moves  up  and  down  along  the  frame  of  the 
apparatus.  The  use  of  the  balance  to  measure  weight  depend-* 
on  the  fact  that  the  spring  obeys  Hooke's  law  closely  for  small 
elongations,  i.  e.,  the  elongation  is  proportional  to  the  change 
in  the  stretching  force.  Thus,  with  the  lower  pan  immersed 
in  water,  we  note  that  the  weight  of  a  certain  mass,  sav  10 
grams,  in  the  upper  pan  produces  an  observed  elongation  of 
the  spring.  If  then  we  substitute  any  solid  for  the  known 
mass  in  the  upper  pan,  giving  an  elongation  not  very  different 
from  the  first,  the  mass  of  this  solid  may  be  readily  computed. 
If  the  solid  is  then  placed  in  the  lower  pan  and  immersed  in 
water,  the  new  elongation  will  give  the  weight  of  the  solid 
in  water ;  or  we  may  add  masses  to  the  upper  pan  until  the 
spring  is  stretched  to  the  same  length  with  the  solid  immersed 
as  when  it  was  in  the  upper  pan.  Then  the  weight  of  the 
masses  we  have  added  gives  the  apparent  loss  of  weight  by 
the  solid  in  water,  or  the  weight  of  the  water  displaced.  The 
mass  added  is  the  mass  of  the  water  displaced,  so  from  this 
and  the  density  of  the  water  the  volume  of  the  solid  can  be 
calculated. 

(a)  By  the  method  outlined  above,  find  the  weight  jn  air 
and  in  water  of  the  solids  furnished.  In  taking  readings  a 
bead  or  point  of  wire  at  the  lower  end  of  the  spring  is  brought 
into  coincidence  with  its  image  in  the  mirror  scale.  Take  care 
that  no  air-bubbles  cling  to  the  lower  pan  or  to  the  solid  when 
immersed.  Take  the  temperature  of  the  water  at  the  beginning 


3]  BOYLE'S  LAW.  5 

ami  at  the  end  of  the  experiment,  use  the  mean,  and  from  the 
Tables  find  the  denisty  of  water  at  this  temperature.  From  the 
data  thus  obtained  calculate  the  density  of  each  solid  in  grams 
per  cc. 

(b)  Find  the  density  of  a  solid  which  floats  in  water.  For 
this  purpose  a  sinker  must  be  used,  but  this  may  be  left  in  the 
lower  pan  throughout  the  experiment. 

(c)  Find  the  density  of  a  salt-solution  by  using     a     solid 
hung  by  a  thread  instead  of  using  the  lower  pan.    Explain  the 
method  used. 

(</)  If  the  temperature  of  the  water  during  the  meas- 
urement in  (a),  taken  on  the  first  solid,  had  been  4O°C.  and 
the  density  of  the  water  had  been  assumed  to  be  i  gram  per 
cc.,  what  percentage  error  would  have  resulted?  If  a  bubble 
of  air  had  been  carried  down  with  the  solid  when  immersed, 

would  the  calculated  density  have  been  greater  or  less?  Why? 

•  -  *  •  •  ^-\ 

3.    BOYLE'S  LAW. 

References. — Watson,   p.    150;    Millikan,    p.    105;    Ames    and    Bliss, 

p.  209. 

The  object  of  this  experiment  is  to  study  the  relation  be- 
tween the  volume  and  the  pressure  of  a  gas  at  a  constant 
temperature,  to  see  how  this  relation  varies  when  the  temper- 
ature is  changed,  and  to  compare  the  behavior  of  two  gases 
when  subjected  to  the  same  pressures  and  temperatures. 
Boyle's  law  for  gases  is  given  by  the  equation,  PV=C,  where 
P  is  the  pressure,  V  the  volume,  and  C  a  quantity  depending 
on  the  temperature,  being  constant  for  a  given  mass  of  the  gas 
if  the  temperature  is  constant.  All  gases  show  some  varia- 
tion from  this  law.  In  the  present  experiment  the 
above  equation  is  to  be  tested,  the  variation  of  C  noted  as  P 
varies  through  a  given  range,  the  amount  of  variation  of  C 
for  two  gases  compared,  and  the  relation  between  the  tem- 


6  BOYLE'S  LAW.  [3 

perature  and  the  values  of  C  for  two  or  more  temperature- 
determined. 

The  apparatus  consists  of  two  closed  tubes,  graduated  and 
mounted  side  by  side,  the  one  containing  air  and  the  other 
carbon  dioxide.  In  each  tube  the  gas  is  confined  over  a  col- 
umn of  mercury  contained  in  a  rubber  tube  connecting  the  two 
closed  tubes  with  an  open  glass  tube  which  can  be  moved  up 
or  down,  thereby  changing  the  pressure  on  the  enclosed  gases. 
The  difference  in  level  of  the  mercury  in  the  open  and  closed 
tubes  may  be  read  from  a  scale  by  using  a  special  metal 
square.  Surrounding  the  closed  tubes  is  a  glass  jacket  in 
which  water  of  a  given  temperature  may  be  placed  in  order 
to  control  the  temperature  of  the  gases. 

(a)  Put  water  at  room  temperature  into  the  jackets.  Take 
a  series  of  settings  for  ten  different  pressures,  some  less  and 
some  greater  than  atmospheric  pressure,  recording  in  tabular 
form  the  corresponding  values  of  the  volumes  and  the  (Differ- 
ences in  level  of  the  two  mercury  menisci  for  each  gas.     The 
settings  for  the  two  gases  are  made  simultaneously,  but  the 
records  for  the  two  should  be  kept  separate. 

(b)  Change  the  temperature  of  the  bath  and   repeat   (a), 
making,  however,  only  five  or  six  settings.     See  that 'the  tem- 
perature remains  constant. 

(c)  Change  the  temperature  again  and  repeat  (b}.     If  ice 
is  available  it  will  be  well  to  take  one  temperature  at  about  o°, 
and  the  other  at  40°  or  5O°C.  (not  higher).    Read  the  barom- 
eter to  obtain  the  atmospheric  pressure. 

(d)  Calculate  the  pressures,  P,  and  the  products,  PV.  cor- 
responding to  the  data  of  (a),  (b),  and  (r),  and  record  in  the 
same  tabular  form  with  those  data. 

From  the  results  for  (a),  plot  a  curve  for  each  of  the  two 
gases  on  the  same  sheet  of  coordinate  paper,  having  volumes 
as  abscissae  and  the  products,  PV,  as  ordinates.  using  any  con- 
venient units  of  measure  along  the  two  axes.  Indicate  the  po- 
rtion of  observation  bv  a  dot  surrounded  bv  a  small  circle. 


4]  THE  vo\ 


/UMENOMETER.  7 

Draw  a  smooth  curve  (not  a  broken  line)  passing  through  or 
near  the  points.  Plot  similar  curves  for  the  other  two  temper- 
atures. Which  gas  follows  Boyle's  law  the  more  closely? 
Calculate  the  greatest  percentage  variation  in  the  value  of  C 
for  any  one  curve.  Write  the  average  values  of  C  for  the 
different  temperatures  for  one  of  the  gases  and  the  correspond- 
ing temperatures,  T,  on  the  absolute  scale.  Do  you  find  any 
definite  relation  between  the  values  of  C  and  T? 

4.    THE  VOLUMENOMETER. 

The  object  of  this  experiment  is  to  find  the  density  of  an 
irregular  solid  by  means  of  the  volumenometer  and  balance.  In 
the  volumenometer  A  is  a  glass  tube  which  may  be  closed  at 
the  top  by  a  ground  glass  plate.  It  corresponds  to  the  closed 
tube  in  the  experiment  on  Boyle's  law.  As  in  that  experiment, 
the  pressure  and  volume  of  the  air  in  A  are  varied  by  raising 
or  lowering  a  tube,  L,  which  is  connected  with  A  through  a 
rubber  tube  containing  mercury.  The  pressure  is  determined 
by  noting  the  difference  in  the  level  of  the  two  mercury  men- 
isci, and  reading  the  barometer.  The  volume  is  unknown. 
The  volume  of  a  portion  of  the  tube  between  two  marks  (a 
and  b),  however,  is  known. 

Let  the  volume  between  a  and  b  be  K,  and  that  above  a  be 
V.  By  determining  the  pressures  when  the  volume  of  air  is 
V  (mercury  meniscus  at  a),  and  again  when  the  volume  is 
V  +  K  (mercury  meniscus  at  b),  an  equation  involving  Boyle's 
law  may  be  written  containing  these  two  volumes  and  the  cor- 
responding pressures.  From  this  equation  V  may  be  calcu- 
lated. The  volume  of  air  in  A  may  be  calculated  in  this  way 
both  with  and  without  the  solid  body  enclosed  whose  volume 
we  desire  to  know.  The  volume  of  this  solid  thus  becomes 
known. 

(a)  With  A  uncovered  bring  the  mercury  meniscus  to  a. 
recording  the  pressure.  Carefully  place  the  plate  on  A,  so  as 


8  DENSITY  OF  AIR.  [5 

to  insure  an  air-tight  joint.  The  plate  must  be  clean  and  have 
on  it  only  a  little  grease.  Lower  the  mercury  meniscus  to  b. 
and  again  read  the  pressure.  Test  for  leakage  by  allowing  the 
tube  to  remain  a  minute  or  more  in  this  position,  and  make 
sure  that  the  heights  of  the  menisci  do  not  change.  First  calcu- 
late K,  and  then,  by  applying  Boyle's  law,  find  V. 

(b)  Remove  the  plate,  place  inside  the  volumenometer  one 
of  the  bodies  whose  density  is  to  be  determined,  and  repeat 
(a).     From  the  volume  V  of  the  air,  found  in  this  case,  and 
the  former  volume  V,  the  volume  of  the  body  is  found.  Weigh 
the  body  and  determine  its  density. 

(c)  Repeat  for  at  least  two  other  bodies. 

(d)  Find  the  densities  of  the  substances  used  as  given  in 
either  Smithsonian  Physical  Tables  or  in  Whiting's  Tables,  and 
accepting  these  values  as  correct,  calculate  the  percentage  error 
of  your  determination  in  each  case. 

What  are  the  advantages  and  the  disadvantages  of  this 
method  for  determining  density?  For  what  kind  of  bodies 
is  it  to  be  recommended? 

5.     DENSITY  OF  AIR. 
Reference. — Millikan,  p.  114. 

Let  a  glass  bulb  of  volume  V  be  weighed  full  of  air  at  at- 
mospheric pressure  Pt,  and  let  M  be  the  mass  necessary  to 
balance  it.  Then  let  the  air  be  pumped  out  until  the  pressure 
is  Po,  the  mass  as  determined  by  weighing  now  being  (M — m), 
where  m  is  the  mass  of  air  that  has  been  pumped  out  between 
the  weighings.  Then  if  d,  and  d2  be  the  densities  correspond- 
ing to  the  pressures  Px  and  P2,  we  have 

(!)         Vdx  —  Vd2  =  m. 

Since  the  reciprocal  of  the  density  is  the  volume  of  unit  mass, 
it  follows  from  Bole's  law  that 


->*>'* 

5]  DSNSITY"OF  AIR. 

Eliminating  d2  from  (i)  and  (2),  *~[  f& 

mP1 

(3)         < 


V(P,  -  Ps) 

In  applying  the  last  equation  it  is  essential  that  the  tempera- 
ture should  be  the  same  during  the  two  weighings.  If  the 
temperature  is  not  the  same,  the  observed  pressure  in  the  sec- 
ond case  must  be  corrected  so  as  to  give  the  pressure  that 
would  have  existed  had  the  temperature  been  the  same  as  dur- 
ing the  first  weighing.  If  Tt  and  T2  are  the  temperatures  (on 
the  absolute  scale)  of  the  air  during  the  first  and  second  weigh- 
ings respectively,  and  P2'  is  the  observed  pressure  in  the  sec- 
ond case,  the  equation  P2/P2'  =  T1/T2,  by  Charles'  law,  gives 
the  corrected  value  of  P2,  the  expansion  of  the  glass  flask 
being  neglected.  The  volume,  V,  is  obtained  by  weighing  the 
bulb  full  of  water  at  a  known  temperature.  The  density,  d0, 
at  o°C.  can  be  obtained  from  dj  by  applying  Charles'  law, 


where  t  is  the  temperature,  centigrade,  of  the  air  used. 

(a)  Carefully  dry  the  flask  by  exhausting  it  several  times 
and   admitting  air   through   a   calcium   chloride   drying  tube. 
Ask  an  assistant  for  instructions  in  regard  to  manipulating 
the  pump.     If  moisture  is  visible  inside  the  flask,  it  may  be 
necessary  to  put  in  a  little  alcohol,  rinse  the  flask,  vaporize 
the  alcohol  over  a  Bunsen  burner,  and  pump  out.     With  the 
dried  flask  in  connection  with  the  manometer  and  the  drying 
tube,  admit  air  at  atmospheric  pressure,  reading  the  pressure. 
Close  the  stop-cock  and  carefully  weigh  the  flask.     Note  the 
temperature. 

(b)  Pump  the  air  out  until  as  low  a  pressure  as  possible 
is  obtained  and  weigh  again  at  this  reduced  pressure.     Again 
note  the  temperature. 

(c)  Fill  the   flask  completely  with   water   up   to  the  stop- 


IO  RELATIVE    DENSITY    OF    CARBON    DIOXIDE.  [6 

cock,  taking  care  to  have  no  water  above  the  stop-cock.  Ask 
an  assistant  to  show  you  how  to  fill  it.  The  temperature  of 
the  water  should  be  recorded  and  its  density  found  from  a 
book  of  Tables.  Dry  the  outside  of  the  flask  and  then  weigh. 
Calculate  the  volume  of  the  flask. 

(d)  Using  the  results  obtained  in  (a),  (b),  and  (c),  find 
the  density  of  the  air,  in  grams  per  cc.,  at  the  given  tempera- 
ture. Calculate  the  density,  d0,  at  o°C.  Take  the  value  given 
in  the  Smithsonian  Tables  and  calculate  your  percentage  error. 
Point  out  the  chief  sources  of  error  in  the  experiment. 

6.     RELATIVE  DENSITY  OF  CARBON  DIOXIDE. 
Reference.  —  Millikan,  p.  114. 

The  relative  density  of  carbon  dioxide  compared  with  air 
as  a  standard  is  to  be  measured.  The  method  employed  is  that 
used  in  Exp.  5.  Using  the  same  symbols  as  there  used,  and 
making  the  weighings  and  noting  the  pressures  as  there  indi- 
cated, we  have  for  the  air, 

mP, 

d.  =  vTPT^p,)' 

If  the  measurements  are  then  repeated  for  the  carbon  dioxide, 

m'P/ 

**  =  WT-^ 

the  symbols  having  the  same  meaning  as  in  the  case  of  air. 
From  (i)  and  (2),  if  D  is  the  relative  density  of  the  carbon 
dioxide, 


~d,  -mP,  (P.'-P,1)' 

from  which  we  see  that  a  determination  of  the  volume  of  the 
flask  is  unnecessary. 

(a)   Read  the  directions  given  under  Exp.  5.    Ask  an  assist- 
ant for  instructions  in  the  use  of  the  pump.     Carefully  dry  the 


7]  THE:  FORCE:  TABLE  n 

flask,  and  fill  it  with  dry  air  admitted  through  the  calcium 
chloride  tube.  Using  a  sensitive  balance,  weigh  the  flask  full 
of  air  at  atmospheric  pressure,  noting  pressure  and  tempera- 
ture. In  weighing  follow  the  method  given  in  Exp.  i. 

(b)  Pump  the  air  out  to  a  low  pressure  and  weigh  the  flask 
again  at  the  reduced  pressure.     If  the  temperature  is  not  the 
same,  within  o°.5,  the  observed  pressure  should  be  corrected 
as  in  Exp.  5. 

(c)  Fill  the  flask  with  dry  carbon  dioxide  at  atmospheric 
pressure.    This  can  best  be  done  by  pumping  out  the  flask  and 
admitting  the  gas  from  the  generator  several  times  in  succes- 
sion.    Take  care  not  to  allow  any  air  to  pass  through  the  acid 
into  the  generator,  and  keep  the  stop-cock  closed   when  not 
using  the  generator.    When  the  flask  is  filled  with  carbon  diox- 
ide at  a  known  pressure  and  temperature,  weigh  it  as  before. 

(d)  Pump  the  carbon  dioxide  out  to  a  low  pressure,  as  in 
the  case  of  the  air,  and  weigh  again. 

(<?)  By  the  use  of  equation  (3)  calculate  from  your  results 
the  relative  density  of  carbon  dioxide  compared  with  air  as 
a  standard.  Using  the  values  of  the  densities  of  air  and  of 
hydrogen  obtained  from  the  Smithsonian  Tables,  calculate  the 
density  (in  gm/cc.)  of  the  carbon  dioxide  and  its  relative  den- 
sity compared  with  hydrogen.  Calculate  the  percentage  error 
of  your  determination  of  the  density  by  comparing  it  with  the 
value  given  in  the  Tables.  What  are  the  principal  sources  of 
error  in  the  experiment?  What  would  be  the  effect  of  having 
a  little  water  in  the  flask  ? 

7.    THE  FORCE  TABLE. 

References. — Millikan,  p.  21;  Watson,  p.  73. 

The  purpose  of  this  experiment  is  to  determine  the  vector 
sum  of  two  forces  acting  on  a  body  in  the  same  plane  and 
along  lines  not  parallel.  Let  the  lines  of  direction  of  the  two 
forces,  fx  and  f2,  intersect  in  a  point,  O,  making  angles,  ax  and 


12  THE  FORCE  TABLE.  [/ 

a2,  with  an  arbitrarily  chosen  axis,  OX.  The  vector  sum  of 
these  forces  is  by  definition  a  force,  f,  given  by  the  diagonal 
of  the  parallelogram  formed  by  fj  and  f2  as  sides.  Let  the  di- 
rectional angle  of  f  be  a.  By  taking  the  projections  of  f,.  f... 
and  f  upon  the  perpendicular  axes,  OX  and  OY,  we  see  by 
construction  that 

1 I )  f  sin  a  =  f ,  sin  ax  -f"  ^2  sm  a2» 

(2)  i   COS   a  =  f t    COS   04  +  f2   COS   a2  ; 

whence,  by  squaring  (i)  and  (2)  and  adding, 

(3)  f2  -  fi2  +  *22  +  2ftf2  C0s(a2  —  a,)  ; 
and,  by  dividing  (i)  by  (2), 

i.  sin  a    -f-  f  sin  a 

(4)         tan  a  = — —     -  . 

1,  cos  a,  -j-  12  cos  a  i 

Equation  (3)  enables  one  to  calculate  the  magnitude  of  the 
resultant  of  the  two  given  forces,  and  by  equation  (4)  the  direc- 
tion of  this  resultant  can  be  determined. 

The  apparatus  used  to  test  these  results  consists  of  an  adjust- 
able iron  table  with  circular  top  graduated  in  degrees.  Pul- 
leys can  be  clamped  to  the  circumference  at  any  chosen  points. 
From  a  pin,  placed  in  a  hole  in  the  center  of  the  table-top, 
three  cords  pass  over  the  pulleys  and  carry  pans  upon  which 
known  masses  are  placed.  The  masses  and  pans  should  be 
weighed  on  the  platform  scales. 

(a)  Arbitrarily  take  f,,  f2,  a1?  a2  as  equal   respectively   to 
(200  -f  m)    gms.   wt.,    (ioo-f-m)   gms  wt.,  35°,  85°,   where 
m  is  the  mass  of  the  pan  holding  the  masses.     Calculate  by 
equations  (3)  and  (4)  the  value  of  f  and  a. 

(b)  Set  one  of  the  pulleys  at  35°  and  one  at  85°.     With 
the  pin  in  place  put  the  requisite  masses  in  the  pans  to  make 
f,  and  f2  equal  to  the  values  chosen  for  them.    Then,  if  a  third 
pulley  be  set  180°  from  the  direction  determined  by  a  as  cal- 
culated in    (a),  and   masses  corresponding  to  the  calculated 


8]     EQUILIBRIUM  CONDITIONS  FOR  THREE  FORCES  AT  A  POINT.    13 

values  of  f  be  added,  the  three  forces  acting  on  the  pin  should 
be  in  equilibrium,  since  the  third  force  is  equal  and  opposite 
to  the  vector  sum  of  ft  and  f2.  Pull  out  the  pin  and  see  wheth- 
er the  calculation  is  correct. 

(c)  In  a  similar  manner  calculate  and  test  two  other  sets 
of  values  chosen  by  you. 

(d)  Select  three  new  sets  of  values  for  f,,  f2,  ax,  and  a2  and 
proceed  as  follows  with  each  set :     Place  a  circular  sheet  of 
manila  paper  on  the  table  and  run  the  pin  through  it.     Set 
the  two  pulleys  at  at  and  a2,  and  place  the  requisite  masses 
on  the  pans.     Mark  with  a  pencil  the  directions  of  the  two 
strings-,  then  remove  the  paper  and  on  the  lines  lay  off  dis- 
tances from  their  intersection  proportional  to  fx  and  f2.     Com- 
plete the  parallelogram  and  determine  from  the  diagonal  the 
value  of  f.     Now  replace  the  paper  on  the  table,  set  the  third 
pulley  opposite  to  f,  and  adjust  the  masses  on  its  pan  to  equal 
the  value  of  f  as  determined  by  the  diagonal.     Pull  out  the 
pin  and  see  if  the  construction  is  correct. 

(e)  Point  out  the  principal  sources  of  error  in  the  two  meth- 
ods used  above. 

If  three  forces  acting  upon  a  body  hold  it  in  equilibrium, 
how  must  their  lines  of  direction  intersect? 

A  ladder  leaning  against  a  smooth  vertical  wall  is  prevented 
from  sliding  by  the  reaction  of  the  ground.  What  forces  are 
acting  on  the  ladder?  Construct  the  line  of  direction  of  the 
reaction  of  the  ground  on  the  ladder. 

8.     EQUILIBRIUM  CONDITIONS   FOR  THREE 
FORCES  AT  A  POINT. 

A  body  acted  on  by  three  forces  will  be  in  equilibrium  if 
their  lines  of  direction  pass  through  the  same  point  and  their 
vector  sum  is  zero.  It  follows  that  the  three  forces  must  lie 
in  the  same  plane,  but  it  is  not  necessary  that  their  points  of 
application  in  the  body  shall  be  the  same  point.  Let  two  of 


14  EQUILIBRIUM  CONDITIONS  FOR  THREE  FORCES  AT  A  POINT.   [8 

the  forces  be  mutually  perpendicular.  The  third  force,  to 
produce  equilibrium,  must  be  equal  and  opposite  to  their  resul- 
tant. If  OA  and  OB  (see  Fig.)  represent  the  two  forces, 
fj  and  f2,  their  resultant  will  be  represented  by  the  diagonal. 


OC',  constructed  upon  OA  and  OB  as  sides.  The  third  force, 
f3,  if  there  is  equilibrium,  will  then  be  represented  by  OC. 
drawn  equal  and  opposite  to  OC'.  As  thus  represented  it  is 
evident  that  the  vector  sum  of  the  three  forces  is  zero.  This 
condition  may  be  expressed  in  the  equations, 

( i )         f  j  —  f  3  cos  a  =  o 


(2) 


L  —  f,  sin  a  =  o. 


These  relations  enable  us,  from  a  knowledge  of  the  magni- 
tudes of  the  two  perpendicular  forces,  to  find  the  magnitude 
and  relative  direction  of  their  resultant ;  or,  from  a  knowledge 
of  the  magnitude  and  direction  (relative  to  two  perpendicular 
lines)  of  a  single  given  force,  to  find  the  magnitudes  of  its 
rectangular  components  in  the  two  lines.  (There  are,  of 
course,  as  many  pairs  of  rectangular  components  of  a  given 
force  as  there  are  pairs  of  mutually  perpendicular  directions.) 
In  the  apparatus  used,  three  masses  (m,,  m,,  and  m3)  are 
suspended  by  cords  passing  over  pulleys  so  that  they  will 
produce  forces  acting  at  a  point,  O.  By  adjusting  the  masses 
and  the  position  of  the  pulleys  the  angle  AOB  is  made  a  right 


9]  UNIFORMLY    ACCELERATED    MOTION.  15 

angle.  For  convenience  of  adjustment,  two  of  the  masses  con- 
sist of  shot  in  buckets.  On  the  paper  fastened  back  of  the 
cords  the  lines  OA,  OB,  and  OC  can  be  traced. 

(a)  First  make  m3  equal  to  500  or  *eoo  gms.     Then  adjust 
m^  and  m2,  by  means  of  shot,  until  all  is  in  equilibrium  with 
the  angle  AOB  equal  to  a  right  angle.     Trace  on  the  paper 
the  directions  of  the  lines  OA,  OB,  and  OC.     Measure  the 
angle  a,  either  by  using  a  protractor,  or  by  a  trigonometric  re- 
lation.   Record  the  value  of  a  and  the  three  forces  ft,  f2,  f3. 

(b)  Repeat  with  a  different  value  of  ms. 

(c)  Choose  definite  values  for  ml  and  m2.     Adjust  m3  so 
that  the  angle  AOB  is  a  right  angle.     Record  the  value  of  the 
forces,  as*d  make  tracings  of  their  directions  as  before.    Repeat 
for  another  set  of  values  of  ml  and  m2. 

(d)  From   the   results   in    (a)    ,    (b),   and    (c)      determine 
whether  the  condition  of  equilibrium  is  satisfied,  first  by  sub- 
stituting the  recorded  values  in  the  equations    (i)    and   (2), 
and  again  by  constructing  the  triangle  of  forces  in  each  case 
and  noting  if  it  is  closed  or  not. 

In  one  case  construct  the  resultant  of  the  forces,  ft  and  f2, 
and  compare  it  in  direction  and  magnitude  with  f3. 

What  do  you  consider  is  the  body,  acted  on  by  the  three 
forces  (fj,  f2,  and  fs),  in  whose  equilibrium  we  are  interested? 

What  are  the  principal  sources  of  error  in  this  experiment? 

What  equations  would  connect  fx,  f,,  and  f3,  if  the  angle 
AOB  were  not  a  right  angle  ? 

9.     UNIFORMLY  ACCELERATED    MOTION. 
References. — Millikan,   p.   9;    Ames   and    Bliss,   p.    70. 

The  purpose  of  the  present  experiment  is  to  determine  the 
acceleration  of  a  freely  falling  body.  Conceive  of  a  body 
moving  in  a  straight  line  with  uniformly  accelerated  motion, 
being  at  a  point  A0  at  a  certain  time,  at  Aj  one  interval  of 
time  later,  at  A0  at  the  end  of  the  second  interval  of  time,  etc. 


1  6  UNIFORMLY  ACCELERATED  MOTION.  [9 

Let  Sj  be  the  distance  covered  during  the  first  interval  of  time, 
s2  the  distance  covered  during  the  second  interval,  etc.  ;  and 
let  t  be  the  number  of  seconds  in  the  given  interval  of  time. 
Let  vx  be  the  average  velocity  of  the  body  during  the  first 
interval,  v2  that  during  the  second  interval,  etc.  Then,  by  the 
definition  of  average  velocity,  we  have, 

(i)         v«=:p      v,==^,etc. 

Let  at  be  the  average  acceleration  between  the  first  and  sec- 

ond intervals  of  time,  a2  that  between  the  second  and  third 

intervals,  etc.  Then,  by  the  definition  of  average  accelera- 
tion, we  have 

v..  —  v,  v,  —  v.. 

(2) 


or  substituting  from  (i), 

•    (3)         a,  =  S-VA     a,  =  5LI1^  etc. 

If  the  body  has  uniformly  accelerated  motion,  alf  a2,  etc.  must 
be  equal. 

(If  the  motion  is  uniformly  accelerated,  the  velocity  will  increase 
at  a  constant  time-rate.  Then  v1?  the  average  velocity  for  the 
first  interval  of  time,  will  be  equal  to  the  instantaneous  velocity  of 
the  body  at  the  middle  instant  of  that  interval;  similarly  v2  will 
be  equal  to  the  instantaneous  velocity  at  the  middle  instant  of  the 
second  interval  of  time,  etc.  Between  the  middle  instants  of  any 
two  successive  time-intervals  the  time  elapsing  is  evidently  equal 
to  t.  The  average  acceleration,  then,  between  the  middle  instants 
of  the  first  and  second  intervals  is  (v.,  —  v^/t,  as  given  above.) 

(a)  The  falling  body  consists  of  a  brass  frame  which  falls 
about  1  20  cm.  along  guides  which  offer  very  little  friction. 
This  frame  carries  with  it  a  tuning  fork,  one  prong  of  which 
carries  with  it  a  stylus  which  traces  a  wavy  line  upon  the 
whitened  glass  plate  clamped  vertically  in  the  support.  The 
release  of  the  fork  by  the  lever  at  the  top  causes  the  prongs 


TO]  THE  FORCE  EQUATION.  17 

to.  vibrate.  A  plumb  line  is  used  to  adjust  the  plate  and  guides 
for  the  fork,  so  that  they  will  be  accurately  vertical.  The 
plate  is  first  covered  with  a  thin  coat  of  bon  ami,  which 
quickly  dries.  It  is  then  placed  in  the  frame  and  adjustments 
made.  At  least  three  good  traces  should  be  obtained. 
A  fine  line  is  next  ruled  along  the  center  of  the  trace ; 
and,  starting  at  any  convenient  point,  points  ten  vibrations 
apart  are  marked  off  and  their  distances  apart,  st,  s2,  etc.,  meas- 
ured. Tabulate  these  values  of  s  and  their  successive  differ- 
ences. Repeat  for  points  twenty  vibrations  apart.  These 
measurements  should  be  made  for  at  least  two  traces. 

(&)  From  the  known  value  of  the  frequency  of  the  fork 
find  t.  Calculate  a  for  each  set  of  observations  and  take  the 
mean.  Is  the  acceleration  constant?  Calculate  the  percentage 
error.  Name  the  principal  sources  of  error. 

10.    THE  FORCE  EQUATION. 

If  P  is  the  resultant  force  acting  on  a  body,  m  its  mass,  and 
1>  the  acceleration  produced,  we  have,  as  the  result  of  defini- 
tion and  experiment, 

( i )         p  =  kmp. 

This  equation  is  called  the  Force  Equation,  or  the  Equation 
of  Motion ;  and  the  purpose  of  the  present  study  is  to  verify 
it  experimentally,  k,  in  the  equation,  is  a  constant  numerical 
factor,  whose  value  depends  upon  the  system  of  units  used. 
This  equation  states  ( I )  that,  if  two  forces  act  on  bodies  of 
the  same  mass,  the  acceleration  produced  will  be  directly  pro- 
portional to  the  forces;  and  (2)  that,  if  two  forces  produce 
the  same  acceleration  in  two  bodies  of  different  mass,  the 
masses  will  be  directly  proportional  to  the  forces.  Let  M,  M 
be  two  equal  masses  suspended  from  a  cord  passing  over  a 
pulley  whose  friction  and  rotational  inertia  we  will  assume  to 
be  negligible.  The  total  mass  suspended  is  2M,  the  resultant 


l8  THK    I'OKCK    EQUATION.  [  1O 

force  acting  on  it  is  zero.  Let  a  mass  ml  be  added  to  one 
side.  The  resultant  force,  P,,  now  is  kn^g,  and  it  will  cause 
the  three  masses  to  move  in  its  direction  with  an  acceleration, 
Pi  ;  hence,  by  equation  (i), 

(2)  P1=k(2M  +  m1)p1. 

If  a  different  force  be  applied  by  replacing  ml  with  a  mass  m2, 
the  resultant  force,  P2,  will  be  km2g,  and  it  will  produce  an 
acceleration,  p2  ;  hence,  by  equation  (i), 

(3)  P2 


Hence  knvg_  P,  _  k(2M  -j    m,)p 

nciicc  -:  —          —  -pr-  —  —  -,  —  —  —  —  ',      or, 

km2g       P.        k(2M  4-  m,)p2 


,^  -f 


m,       (2M  -f-  m,)P; 

An  experimental  verification  of  equation  (4)  will  constitute 
a  verification  of  equation  (  i  )  ,  though  it  will  not,  of  course', 
determine  the  value  of  the  constant,  k. 

The  apparatus  used  in  Exp.  9  is  employed,  with  the  addi- 
tion of  a  pulley-attachment  at  the  top  over  which  a  cord  passes, 
from  one  end  of  which  the  fork  is  suspended  and  from  the 
other  end  a  number  of  masses  just  sufficient  to  balance  the 
fork  and  the  friction  of  the  pulley.  Note  the  precautions  given 
in  Exp.  9.  Special  care  should  be  taken  to  insure  as  little  fric- 
tion as  possible. 

(a)  Adjust  the  apparatus  so  that  a  good  trace     may     be 
obtained  and  so  that  a  slight  tap  will  cause  the  fork  to  descend 
without  acceleration.     The  forces,  including  friction,  are  then 
just  balanced.      Whiten   the   plate   with   the  preparation    fur- 
nished. 

(b)  Remove  a  mass,  m,,  from  the  balancing  weights.  Note 
the  total  mass,  M,,  of  the  moving  system.     Obtain  two  good 
traces. 

(c)  Repeat  with  a  different  mass,  m2,  removed,  the  total 
mass  of  the  system  now  being  M2. 


CT    L 

0 


Il]  THE    SIMPLE    PENDULUM.  19 

•(d)   Repeat  again  with  a  third  mass  removed. 

(e)  Measure  the  traces  as  explained  in  Exp.  9,  using  ten 
vibrations  of  the  fork  as  the  interval  of  time.  Calculate  the 
accelerations  plt  p2,  p?>,  corresponding  to  (&),  (c),  (d)  above. 
Then,  by  equation  (4),  we  should  have  m^/m2  =  M^/M-gp., 
and  m,/m3  =  M^/M^.  Test  this,  calculating  the  percent- 
age error  in  each  case. 

Name  the  principal  sources  of  error  in  the  experiment,  and 
account  for  any  discrepancies  in  your  results. 

IT.     THE  SIMPLE  PENDULUM. 
References. — Ferry    and    Jones,    p.    66;    Millikan,    p.    95. 

For  vibrations  of  small  amplitude  the  period  of  a  simple 
pendulum  is  given  by  the  equation, 

T=  2 

where  T  is  the  time  of  one  complete  vibration,  1  is  the  length 
of  the  pendulum,  and  g  is  the  acceleration  due  to  weight.  If 
T  and  1  are  known  for  any  place,  g  can  be  determined  for  that 
place. 

In  the  present  experiment  T  is  to  be  measured  by  compar- 
ing by  the  "method  of  coincidences"  the  period  of  the  simple 
pendulum  with  that  of  a  clock  pendulum  of  known  period.  An 
electric  circuit  is  completed  through  an  electric  bell,  the  clock 
pendulum,  the  simple  pendulum,  and  the  mercury  contacts  at 
the  bottom  of  each  pendulum.  Assume  that  the  period  of  the 
clock  pendulum  is  two  seconds,  that  is,  that  the  time  of  a  single 
swing  or  half-vibration  is  one  second.  If  the  period  of  the 
simple  pendulum  were  the  same  and  the  two  pendulums  be 
started  together,  they  would  vibrate  in  coincidence  and  the 
bell  would  ring  with  every  passage.  If,  however,  the  time  of 
a  single  swing  of  the  simple  pendulum  were  less  than  one 
second,  say  by  i/n  th  of  a  second,  it  would  gain  on  the  clock 


2O  TUK    SlMl'LK     1'KMH'l.r.M.  [ll 

pendulum  and  thus  fall  out  of  coincidence  with  it,  so  that 
the  bell  would  cease  to  ring  until  n  seconds  later,  when  the 
two  pendulums  would  be  in  coincidence  again.  Let  us  sup- 
pose that  the  time  between  these  successive  coincidences  is 
100  seconds,  then  we  know  that  in  this  time  the  clock  pen- 
dulum has  made  one  hundred  half-vibrations  and  the  simple 
pendulum  one  more,  or  101  half-vibrations.  In  other  words, 
the  simple  pendulum  has  made  101  half-vibrations  in  100  sec- 
onds, hence  the  value  of  its  half-period  is  100/101  seconds. 
If,  on  the  other  hand,  the  simple  pendulum  had  been  observed 
to  fall  behind  the  clock  pendulum,  and  the  time  between  suc- 
cessive coincidences  remained  the  same,  we  would  know  that 
its  half-period  is  100/99  seconds. 

(a)  The  simple  pendulum  used  consists  of  a  brass  sphere 
suspended  from  a  knife-edge  by  a  wire  so  that  the  length  is 
adjustable.  The  mercury  contact  below  should  be  so  adjusted 
that  the  platinum  point  on  the  ball  passes  freely  through  it. 
Adjust  the  pendulum  so  that  its  length  is  either  greater  or  less, 
by  2  or  3  cm.,  than  that  of  a  pendulum  beating  seconds.  Two 
different  lengths  (in  successive  determinations)  should  be 
used  such  that  one  is  greater  and  the  other  less  than  that  of 
a  pendulum  beating  seconds.  In  getting  the  length  it  is  well 
to  measure  with  a  meter  rod  and  square  to  the  top  of  the  ball, 
and  then  to  determine  the  diameter  of  the  ball  with  the  cali- 
pers. After  adjusting,  start  the  ball  swinging  in  an  arc  of 
about  10  cm.,  and  record,  from  the  clock  in  minutes  and  sec- 
onds, the  times  of  six  successive  coincidences  between  the 
simple  pendulum  and  the  clock  pendulum.  If  the  bell  rings 
for  more  than  one  swing  during  each  coincidence,  take  the 
mean  of  the  times  of  the  first  and  last  rings  as  the 
time  of  the  coincidence.  Find  the  difference  in  time 
between  the  first  and  fourth  coincidences,  the  sec- 
ond and  fifth,  the  third  and  sixth,  and  take  the 
mean.  From  this  calculate  the  period.  Be  sure  to  note 
whether  the  pendulum  was  gaining  or  losing  on  the  clock. 


12]  THI$  PRINCIPLE  OF  MOMENTS.  21 

Calculate  the  value  of  "g"  for  Berkeley  for  the  two  cases  and 
take  the  mean. 

(&)  What  effect  would  be  produced  upon  the  vibration  of 
a  pendulum  by  carrying  it,  (i)  to  a  mountain  top,  (2)  from 
the  equator  to  the  pole  of  the  earth?  In  what  way  does  the 
pendulum  used  in  this  experiment  fall  short  of  the  require- 
ments for  a  simple  pendulum?  What  is  the  object  of  taking 
a  small  amplitude  of  vibration  ? 


12.    THE  PRINCIPLE  OF  MOMENTS. 

References. — Millikan,  p.  29;  Ames  and  Bliss,  p.  118. 

The  purpose  of  this  experiment  is  to  determine  the  condi- 
tion which  must  be  satisfied  if  a  body,  acted  upon  by  three 
or  more  forces  in  the  same  plane,  is  to  remain  in  equilibrium 
with  reference  to  rotation.  In  order  that  zrbody  at  rest  shall 
remain  at  rest,  or  a  body  in  motion  remain  in  motion  with  con- 
stant linear  and  angular  velocity,  the  vector  sum  of  all  of  the 
forces  acting  upon  it  must  be  zero  and  the  algebraic  sum  of 
the  moments  of  these  forces  about  any  axis  must  be  zero.  In 
the  case  where  all  the  forces  are  in  the  same  plane,  the  second 
of  these  conditions,  sometimes  called  the  Principle  of  Moments, 
requires  that  the  sum  of  the  moments  of  all  the  forces  about 
any  point  in  the  plane  selected  as  a  center  of  moments  shall 
be  zero.  To  prove  this  it  is  only  necessary  to  show  that  the 
sum  is  zero  for  one  selected  point,  provided  that  this  point  is 
so  chosen  as  not  to  lie  in  the  line  of  any  of  the  forces.  (The 
proof  of  this  for  the  case  where  there  are  three  forces  is  left 
to  the  student.)  If  a  point  in  the  line  of  any  force  were  chosen, 
the  moment  of  that  force  with  reference  to  that  point  would 
be  zero  no  matter  what  the  value  of  the  force ;  hence,  the  result 
would  not  be  a  test  of  the  principle. 

The  apparatus  used  to  test  the  principle  consists  of  a  circu- 
lar table  with  a  movable  disk  resting  on  bicycle  balls.  The 


22  THE  PRINCIPLE  OF   MOMENTS.  [l2 

disk  may  be  pivoted  in  the  center  if  desired.  To  pegs,  placed 
at  will  in  the  disk,  cords  are  attached  which  pass  over  pulleys 
clamped  at  different  points  around  the  circular  table.  From 
the  ends  of  the  cords  are  suspended  known  masses  whose 
weight  produces  the  forces  required. 

(a)  Pivot  the  disk  in  the  center  and  place  a  sheet  of  manila 
paper  upon  it.     Attach  cords  to  the  disk  at  three  different 
points  chosen  at  random ;  and,  placing  the  pulleys  at  any  con*- 
venient  points,  add  masses  until  the  three  forces  are  of  conven- 
ient values.     See  that  the  disk  is  free  to  move  on  the  bicycle 
balls ;  then  mark  points  or  lines  on  the  paper  to  indicate  the 
directions  of  the  forces.     Note  the  magnitude  of  the  forces, 
counting  in  the  weight  of  the  pan  in  each  force. 

(b)  Remove  the  paper,  trace  the  lines  of  direction  of  the 
forces,  and   make  the  measurements   necessary  to  determine 
their  moments  about  the  pivot  as  an  axis.     Find  the  sum  of 
the  moments,  taking  those  as  positive  which  tend  to  produce 
a  counter-clockwise  rotation  about  the  given  axis  and  those 
as  negative  which  tend  to  produce  a  clockwise  rotation. 

(c)  Choose  any  arbitrary  point  on  the  paper  used  in   (a) 
and   (b),  and  find  the  sum  of  the  moments  about  this  point 
as  a  center  of  moments.  Why  is  not  the  sum  zero? 

(d)  Remove   the   pivot,    and    repeat     (a)    and    (b)    once, 
selecting  in  turn  as  centers  three  points  as  widely  separated 
as  possible.     Find  the  sum  of  the  moments  as  before.     Also 
find  the  vector  sum  of  the  forces  by  the  method  of  the  closed 
polygon. 

(e)  Repeat  (d),  using  four  forces  instead  of  three. 

(/)  From  the  data  of  (a)  and  (b)  determine  the  vector 
sum  of  the  three  forces  used  in  that  case.  If  this  sum  is  not 
zero,  it  means  that  the  pivot  itself  exerted  a  force  on  the  disk 
in  the  same  plane  with  the  three  forces.  What  do  you  con- 
clude is  the  magnitude  and  direction  of  this  force?  Draw- 
its  line  of  direction  on  the  paper,  and  then  repeat  (c),  includ- 


13]  THE    MODEL,   RALANCE.  23 

ing  now  in  your  sum  the  moment  of  the  force  clue  to  the  pivot. 
Is  the  sum  now  approximately  zero? 

(g)  In  the  various  cases  of  equilibrium  considered  above, 
what  do  you  find  the  vector  sum  of  the  forces  to  be?  What 
have  you  found  to  hold  true  for  the  moments  of  these  forces? 
Calculate  the  percentage  error  for  one  case. 

13.     THE  MODEL  BALANCE. 
Reference. — Glazebrook  and  Shaw,  p.  83. 

A  model  balance  is  a  simplified  beam  balance  used  to  test 
the  relation  between  the  sensitiveness  of  the  balance  and  its 
dimensions  and  load.  By  "sensitiveness"  is  meant  the  facil- 
ity with  which  the  pointer  of  tthe  balance  can  be  deflected 
when  there  is  a  small  difference  between  the  masses  suspended 
from  the  two  sides  of  the  beam.  The  sensitiveness  of  the  bal- 
ance depends  upon  the  length  and  mass  of  the  beam,  the 
load  in  the  pans,  the  distance  between  tRC^enter  of  weight  of 
the  beam  and  the  central  supporting  knife-edge,  and  upon 
whether  the  beam  is  straight  or  curved  up  or  down.  To  obtain 
an  expression  showing  the  character  of  this  dependence  we 
need  to  apply  the  principle  of  moments. 
Let  m  =  the  mass  of  the  beam, 

1  =  the  length  of  the  beam-arm    (the  two  being  as- 
sumed equal), 
M  =  the  mass  hung  on  each  side,  including  the  mass 

of  the  scale-pan, 
h  =  the   distance   from  the  central  knife-edge  to  the 

center  of  weight  of  the  beam, 
x  =  a  small  excess  mass  placed  in  one  pan, 
a  =  the  deflection  produced  by  the  addition  of  x, 
ft  =  the  angle,  for  the  given  load,  between  a  horizon- 
tal line  and  the  line  drawn  from  the  central  knife-edge  to  the 
knife-edge  at  either  end,  when  the  beam  is  so  placed  that  the 
two  angles  which  can  be  thus  formed  are  equal,     ft  will  be 


24  THE    MODEL   BALANCE.  [13 

positive  if  the  beam  is  concave  upwards,  negative  if  the  beam 
is  concave  downwards.  Applying1  the  principle  of  moments 
for  the  case  of  equilibrium,  the  central  knife-edge  being  the 
center  of  moments,  we  have 

(l)       (M-fx)gl  COS   (£  — a)— MglcOS    (£  +  a) 

—  mgh  sin  a  =  o. 
Expanding,  collecting  terms,  and  transposing, 

[mh  —  (2M  -{-  x)  sin  ft]  sin  a  =  Ix  cos  (3  cos  a,  or 

(->\  tan  a 1  cos  /S 

~x~    "  mh  —  (2M  +  x)  1  sin  0' 

If  the  beam  is  straight,  ft  =  o  and 

tan  a  _     ]_ 

x       "  mh' 

The  sensitiveness  is  measured  by  the  ratio,  tan  o/x.  Hence, 
in  the  case  of  a  straight  beam,  it  is  independent  of  the  load 
and  increases  with  any  arrangement  which  makes  the  fraction, 
1/mh,  larger.  In  the  case  of  a  curved  beam  the  sensitiveness 
is  dependent  upon  the  load  and  also  upon  the  extent  and 
direction  of  the  curvature. 

The  model  balance  provided  allows  ample  modification  of 
the  several  quantities  in  equation  (2).  The  following  possi- 
bilities are  at  once  apparent :  ( i )  the  center  of  weight  of  the 
beam  may  be  raised  or  lowered  according  as  the  central  knife- 
edge  is  placed  in  the  lower  or  upper  hole  in  the  beam;  (2)  the 
length  of  the  beam  may  be  varied  by  hanging  the  masses  at 
different  distances  from  the  center;  (3)  the  mass  of  the  beam 
may  be  increased  by  inserting  a  brass  cylinder  in  the  hole 
which  marks  the  center  of  weight  of  the  beam ;  (4)  the  points 
of  application  of  the  masses  may  be  placed  level  with,  above, 
or  below  the  central  knife-edge,  thus  making  the  beam  straight, 
or  curved  up  or  down;  (5)  the  ratio  of  the  lengths  of  the 
beam-arms  may  be  varied. 


13]  THE    MODEL   BALANCE.  25 

(a)  Starting  with  the  knife-edge  in  the  upper  hole  of  the 
beam,  adjust  so  that  the  pointer  hangs  at  the  middle  of  the 
scale.     Hang  successively  several  one-gram  masses  from  the 

peg  at  one  end  of  the  upper  row,  noting  if  the  deflection  is    -jM  / 
approximately   proportional    to   the   number   of   masses    used,  dr  y 
Why  should  the  deflection  not  be  strictly  proportional  to  the 
number  of  masses  ?    Remove  the  masses  and  hang  an  unknown 
mass  in  their  place.     From  the  deflection  produced,  calculate 
the  mass  of  the  unknown.      V^>?* 

(b)  Hang  the  one-gram  masses  from  the  inner  peg  of  the 
upper  row.     Compare  the  results  with  those  of  (a),  and  state 
how  the  sensitiveness  depends  upon  the  length  of  the  beam- 
arm,  other  conditions  remaining  the  same. 

(c)  Place  the  brass   cylinder   in  position   at  the  center  of 
weight  of  the  beam,  and  thus  increase  its  mass.     Determine 
how  this  affects  the  sensitiveness. 

(d)  Remove  the  brass  cylinder  and  change  the  knife-edge 
to  the  lower  hole  of  the  beam.     Test  the  sensitiveness  and 
compare  with  (a).     How  does  the  sensitiveness  depend  upon 
the  position  of  the  center  of  weight  of  the  beam  with  refer^ 
ence  to  the  central  knife-edge? 

(e)  Change  the  knife-edge  back  to  the  upper  hole  of  the 
beam.     Hang  a  5o-gram  mass  on  each  end  peg  of  the  upper 
row.     Test  the  sensitiveness  and  compare  with  (a).  Does  the 
sensitiveness  for  this  form  of  beam  change  with  the  load.     If 
so,  account  for  it. 

(/)  Transfer  the  5<>gram  masses  to  the  end  pegs  of  the 
lower  row,  and  test  the  sensitiveness.  Compare  with  (e). 
How  is  the  sensitiveness  affected  when  the  beam  is  curved 
down? 

(g)  Repeat  (/)  with  the  5o-gram  masses  removed.  Com- 
pare with  (a).  When  the  load  is  small  what  effect  has  the 
curvature  on  the  sensitiveness? 

A  sensitive  chemical  balance  is  usually  made  with  the  beam 
curved  slightly  upwards  when  there  is  no  load  in  the  pans  A 


26  SrKI'.U'K    TK.XSIOX     r,V    DIRECT     MKASrkKMF.XT.  [14 

medium  load  straightens  the  beam  and  an  excess  load  causes 
a  downward  curvature.  From  the  results  above  show  how 
the  sensitiveness  changes  with  the  load  on  account  of  the 
curvature  of  the  beam.  Illustrate  by  a  diagram,  applying  the 
principle  of  moments  to  explain  the  action. 

14.     SURFACE  TENSION   BY  DIRECT   MEASURE- 
MENT. 

References. — Watson,   p.    189;   Hastings  and   Beach,   p.    138;    Milli- 

kan,  p.  195. 

The  method  employed  will  be  that  of  making  the  force  due 
to  surface  tension  in  a  given  case  the  equilibrant  of  a  known 
force.  A  wire  rectangle  is  hung  from  the  spring  of  a  Jolly's 
balance  and  allowed  to  dip  in  a  soap  solution  which  forms  a 
film  across  the  rectangle.  When  equilibrium  is  established  the 
force  due  to  surface  tension  in  the  two  surfaces  of  the  film 
must  just  balance  the  tension  in  the  spring.  By  knowing  the 
force  which  will  stretch  the  spring  the  same  amount  we  have 
a  measure  of  the  product  of  surface  tension  (which1  is  the 
force  per  centimeter  width  exerted  by  the  surface  film)  and 
twice  the  width  of  the  rectangle  where  it  is  cut  by  the  surface 
of  the  solution.  If  T  is  the  value  of  the  surface  tension,  1 
the  width  of  the  rectangle  along  the  surface  of  the  liquid,  and 
F  the  force  exerted  by  the  spring,  write  the  equation  giving 
the  value  of  T. 

The  Jolly's  balance  used  is  a  very  sensitive  one,  and  must 
be  handled  with  great  care.  Ask  for  directions  if  its  opera- 
tion is  not  already  understood.  The  catch  above  the  pan  of 
the  balance  allows  only  a  small  motion  of  the  lower  end  of  the 
spring,  the  extension  of  the  latter  being  produced  by  raising 
the  upper  end  of  the  spring  by  means  of  a  telescoping  tube 
moved  by  rack  and  pinion,  the  scale  and  vernier  on  this  tube 
measuring  the  extension.  To  make  a  measurement  the  clamp 
holding  the  catch  is  raised  or  lowered  (the  platform  holding 
the  beaker  being  adjusted  at  the  same  time)  until  the  spring 


14]  SURFACE   TENSION    BY  DIRECT    MEASUREMENT.  27 

almost  supports  the  weight,  after  which  an  exact  setting  is 
made  by  means  of  the  telescoping  tube.  If  any  change  is  then 
made  in  the  pull  on  the  end  of  the  spring,  this  change  is 
measured  by  the  amount  the  spring  must  be  shortened  or 
lengthened  by  the  telescoping  tube  to  restore  the  equilibrium. 
Wire  rectangles  of  different  sizes  and  a  wide  beaker  are  pro- 
vided. The  greatest  care  must  be  taken  that  the  beaker  and 
rectangles  are  clean.  They  should  be  washed  in  caustic 
potash  and  rinsed  thoroughly  in  hot  water  before  being  used 
and  before  changing  to  another  liquid.  Do  not  touch  the 
inside  of  the  beaker,  the  liquid,  or  the  part  of  the  rectangle 
on  which  the  film  is  formed. 

(a)  Suspend  a  rectangle,  2  cm.  wide,  from  the  spring,  and 
let  it  be  partially  immersed  in  a  beaker  of  soap  solution.  Read 
the  extension  of  the  spring  when  there  is  no  film  in  the  rect- 
angle, and  again  with  a  film  across  it.    Take  three  sets  of  read- 
ings.    Note  whether  the  pull  of  the  film  depends  on  the  depth 
to  which  it  is  immersed.     If  it  does  depend  on  the  depth,  be 
careful  to  have  the  same  depths  in  all  measurements. 

Repeat  these  measurements  using  rectangles  4  cm.   and  6 
cm.  wide. 

(b)  Calibrate  the  balance  by  observing  the  extension  pro- 
duced by  known  masses. 

(c)  Use  the  rectangle  4  cm.   wide,   cleaning  it     and     the 
beaker  thoroughly,  and  repeat  (a)  with  water  fresh  from  the 
tap.     As  no  film  will  form  with  pure  water,  take  the  reading 
of  the  balance  when  the  under  side  of  the  upper  wire  of  the 
rectangle  is  just  above  the  surface  of  the  water  and  not  in 
contact  with  it ;  and  again,  after  immersing  the  upper  wire  of 
the  rectangle  so  as  to  wet  it,  take  a  reading  when  it  breaks 
away  from  the  surface.     Take  three  sets  of  readings. 

(rf)   Repeat  (c),  using  water  to  50° C.  or  higher. 
(e)   Repeat  (c),  using  alcohol. 

(/)   From  the  data  taken  in  (a),  state  how  the  total  tension 
in  the  film  varies  with  its  width.     Calculate  the  surface  ter- 


28  CAPILLARITY.      RISE  OF  LIQUIDS  IN  TUBES.  [15 

sion,  T,  in  dynes  per  cm.,  for  the  liquids  used  in  (a),  (c), 
(d)t  and  (e),  comparing  the  values  obtained  and  pointing  out 
how  the  surface  tension  is  affected  by  the  temperature. 

15.    CAPILLARITY.     RISE  OF  LIQUIDS  IN  TUBES. 

References. — Watson,   p.    194;   Watson's  Practical   Physics,   p.    139; 
Millikan,  p.  194;  Miller,  p.  113. 

In  the  present  experiment  the  values  of  the  surface  tension 
of  water  and  of  alcohol  are  to  be  measured  by  observing  the 
rise  of  these  liquids  in  capillary  tubes.  When  the  inner  sur- 
face of  a  tube  is  wet  by  a  liquid,  the  surface  tension  of  the 
latter  may  be  considered  as  acting  vertically  upward  at  all 
points  around  the  circumference  of  the  tube.  The  total  upward 
force  is  then  27rrT,  where  r  is  the  radius  of  the  tube  and  T 
the  surface  tension.  If  the  tube  is  of  small  bore,  the  liquid 
will  rise  inside  the  tube,  equilibrium  being  established  "when 
the  weight  of  the  liquid  within  the  tube  above  the  level  of 
the  liquid  outside  equals  the  total  upward  force  due  to  surface 
tension.  If  d  is  the  density  of  the  liquid,  h  its  height  in  the 
capillary  tube  above  the  surface-level,  write  the  expression 
which  gives  the  weight  (in  dynes)  of  the  column  of  liquid. 
Equate  this  expression  to  2?rrT.  From  the  equation  thus 
formed  T,  the  surface  tension  in  dynes  per  centimeter,  can 
be  found. 

(a)  Capillary  tubes  of  different  sizes  are  provided.  These 
may  be  thermometer  tubes  or  larger  glass  tubing  drawn  out 
to  a  fine  bore.  In  either  case  every  precaution  must  be  taken 
to  have  the  tubes  perfectly  clean  and  free  from  all  traces  of 
grease.  They  may  be  cleaned  with  caustic  potash  solution, 
then  rinsed  with  tap  water  and  dried  by  drawing  a  stream  of 
air  through  them  with  the  jet-pump.  With  a  rubber  band 
fasten  the  tubes  side  by  side  to  a  glass  scale,  and  place  the 
scale  and  tubes  vertically  in  a  small  dish  of  distilled  water. 
Lower  the  tubes  first  to  the  bottom  of  the  dish  so  as  to  wet 


l6]  RISE    OF    LIQUIDS    BETWEEN    PLATES.  29 

the  inside  for  some  distance  above  the  point  to  which  the 
water  will  rise.  Then  clamp  them  with  the  ends  below  the 
surface,  and  note  on  the  scale  the  point  to  which  the  water 
rises  in  each  tube.  To  obtain  the  reading  for  the  water  sur- 
face in  the  dish  a  wire  hook  is  provided,  which  should  be 
brought  up  so  that  the  point  is  just  even  with  the  surface. 
Then  read  the  height  of  this  point  on  the  glass  scale. 

(b)  Measure  the  inside  diameter  of  the  tube  with  a  microm- 
eter microscope.     If   drawn-out   tubing  is   used,   scratch   the 
tube  with  a  file  at  the  point  to  which  the  water  rises,  break 
it  and  measure  the  diameter  of  the  end.    Calculate  the  surface 
tension  of  water  and  compare  this  value  with  the  value  found 
in  Experiment  14. 

(c)  In  the  same  way  find  the  surface  tension  of  alcohol. 

(d)  From  the  data  taken  in  (a)  calculate  the  difference  in 
pressure  on  the  two  sides  of  the  surface  film.     Does  this  dif- 
ference in  pressure,  taken  in  connection  with  class-room  work 
or  reading,  suggest  a  relation  other  than  that  used  in  (a)  for 
finding  the  surface  tension.     If  so,  calculate  the  surface  ten- 
sion of  water  by  this  method. 

Would  the  water  rise  as  high  in  the  tubes  had  the  experi- 
ment been  performed  in  a  'Vacuum"?  Explain. 

16.     RISE  OF  LIQUIDS  BETWEEN  PLATES. 

Reference. — Hastings   and   Beach,   p.   146. 

.   ;.;•      i .;  . .;   •';•.'• 

In  the  present  experiment  the  surface  tension  of  water  and 
of  alcohol  is  to  be  measured  by  means  of  the  rise  of  the  liquid 
in  a  wedge-shaped  space  between  two  plates  of  glass. 

Two  plates  of  glass  are  separated  by  two  thin  pieces  of  brass 
placed  between  the  edges  at  one  side,  and  a  single  thicker 
piece  on  the.  opposite  side.  The  plates  are  clamped  together 
and  placed  upright  in  a  shallow  dish  of  liquid.  If  the  liquid 
wets  the  plates,  it  will  rise  in  the  wedge-shaped  space.  The 
general  effect  is  similar  to  that  obtained  by  a  row  of  small 


30  RISE   OF   LIQUIDS   BETWEEN    PLATES.  [l6 

tubes  of  gradually  decreasing  bore.  We  may  consider  any 
very  small  rectangular  prism  of  the  liquid  of  length  x,  thick- 
ness d  (the  distance  between  the  plates  at  the  point  chosen), 
and  height  h.  Assuming  the  surface  tension  to  act  vertically 
upward  where  the  liquid  wets  the  plates,  the  whole  upward 
pull  along  the  two  edges  of  the  prism  will  be  2Tx,  where  T 
is  the  surface  tension.  This  force  must  equal  the  weight  of 
the  prism  of  the  liquid,  which  is  hxdDg,  where  D  is  the  density 
of  the  liquid.  From  this  relation  T  can  be  found. 

(a)  Clean  the  plates  very  carefully  with  caustic  potash 
solution,  and  rinse  with  water.  Clamp  them  together  as  indi- 
cated above,  and  upon  one  side  of  one  of  the  plates  place  a  thin 
sheet  of  white  paper.  Stand  the  plates  upright  in  a  shallow 
vessel  of  distilled  water,  and  looking  through  the  paper  and 
the  plates  toward  the  light,  trace  the  surface  of  the  liquid 
between  the  plates,  the  surface  of  the  liquid  in  the  dish,  and 
the  outlines  of  the  three  pieces  of  metal.  Removing  the  sheet 
of  paper,  draw  a  line  on  the  paper  through  the  position  of  the 
outer  edge  of  the  two  thin  pieces  of  metal.  This  line  should 
be  perpendicular  to  the  line  representing  the  surface  of  the 
liquid  in  the  dish.  Also  draw  a  line,  parallel  to  the  first, 
through  the  inner  edge  of  the  thicker  piece  of  metal.  Select 
any  point,  P,  on  the  curve  representing  the  surface  of  the 
liquid  between  the  plates.  From  this  point  draw  a  line  per- 
pendicular to  the  line  representing  the  surface  of  the  water  in 
the  dish,  and  call  its  length  h.  Draw  another  line  through  P 
perpendicular  to  the  first  line,  and  let  the  length  along  this 
line  from  P  to  the  line  which  coincides  with  the  outside  edge 
of  the  two  thin  metal  pieces  be  1.  Let  the  whole  distance  be- 
tween the  lines  through  the  edges  of  the  metal  pieces  be  L. 
Measure  the  thickness  of  these  metal  pieces  with  a  micrometer 
caliper,  calling  the  thickness  of  the  inner  ones  dlt  and  that 
of  the  thicker  one  d2.  Then  at  the  point  P, 


I1/]  PRESSURE  AND  RADIUS  OF  A  SOAP-BUBBLE.  3! 

Derive  this  equation.  From  the  values  of  d  and  h  thus  found, 
calculate  the  surface  tension  of  water.  Repeat  the  measure- 
ments and  calculation  for  one  or  two  other  points  on  the 
curve. 

(b)   Repeat   (a),  using  alcohol  instead  of  water,  and  find 
the  surface  tension  of  alcohol. 


17.     PRESSURE  AND  RADIUS  OF  A  SOAP-BUBBLE. 
References. — Watson,   p.    193;   Watson's    Practical    Physics,   p.    143. 

The  purpose  of  this  experiment  is  to  determine  how  the 
excess  pressure  inside  of  a  soap-bubble  depends  upon  its  ra- 
dius. The  surface  tension  in  the  outside  and  inside  surfaces  of 
a  soap-bubble  tends  to  contract  it,  and  does  contract  it  until 
this  force  is  counterbalanced  by  the  excess  pressure  of  the 
compressed  air  within  the  bubble.  To  determine  the  relation 
between  this  excess  pressure  and  the  radius  of  the  bubble,  let 
us  consider  that  a  weightless,  air-tight,  rigid  surface  is  made 
to  pass  horizontally  through  the  bubble  so  as  to  divide  it  into 
two  hemispheres  without  changing  the  pressure,  the  film  of 
each  hemisphere  attaching  itself  to  this  surface  all  the  way 
around.  Let  the  lower  of  these  hemispheres  now  be  removed, 
leaving  the  other  undisturbed. 

Consider  the  forces  acting  on  the  bottom  of  the  undisturbed 
hemisphere.  The  lower  side  of  this  plane  is  acted  upon  by  an 
upward  force  of  P-n-r2  dynes,  where  P  is  the  atmospheric  pres- 
sure in  dynes  per  sq.  cm.,  and  r  is  the  radius  of  the  hemisphere. 
The  upper  side  of  the  same  plane  is  acted  upon  by  two  forces, 
one  downward  and  due  to  the  inside  air-pressure,  the  other 
upward  and  due  to  the  tension  in  the  soap-bubble  film ;  the  first 
is  equal  to  (P  -f  p)^1"2  dynes,  where  p  is  the  excess  pressure 
inside  the  bubble  above  the  value  of  the  air-pressure  outside ;  the 
second  is  equal  to  2T(27rr)  dynes,  where  T  is  the  value  of  the 
surface  tension  per  centimeter  length  of  the  circumference. 


32  PRESSURE  AND  RADIUS  OF  A   SOAP-BUBBLE.  [if 

Since  there  is  equilibrium  the  algebraic  sum  of  these  forces  is 
zero;  hence, 

PTrr2  +  47rrT  —  (  P  +  p)7rr2  =  o. 
From  which 


This  shows  that  the  excess  pressure  within  the  bubble  is  in- 
versely proportional  to  its  radius. 

The  apparatus  consists  of  two  essential  parts  (see  Watson's 
Practical  Physics,  Fig.  58)  :  the  first  is  an  enclosed  box  with 
dish  of  soap  solution,  and  the  tube  for  blowing  the  bubble,  with 
a  mirror  and  movable  thread  (on  graduated  scale)  for  meas- 
uring the  diameter  of  the  bubble;  the  second  is  the  delicate 
pressure-gauge.  This  last  consists  essentially  of  a  sort  of 
U-tube  containing  a  light  liquid  and  having  arms  at  almost 
1  80°  to  each  other.  One  arm  is  always  open  to  the  air,  while 
the  other  may  be  placed  in  communication  with  the  air  or  with 
the  inside  of  the  soap-bubble.  A  very  small  difference  in  pres- 
sure may  cause  a  considerable  motion  of  the  liquid.  A  scale 
on  the  board  on  which  it  is  mounted  enables  one  to  read  the 
position  of  the  two  ends.  With  both  ends  of  the  tube  open  to 
the  air  the  position  of  these  ends  is  read,  then  one  end  is 
placed  in  communication  with  the  bubble.  A  micrometer  screw 
at  the  end  may  now  be  moved  so  as  to  tilt  the  board  on  which 
the  tube  is  mounted  and  bring  the  ends  of  the  liquid  column 
back  to  the  same  points  on  the  scale.  If  H  is  the  vertical  dis- 
tance through  which  the  micrometer  screw  has  been  moved 
to  do  this,  L  the  distance  from  the  screw  to  the  hinge  of  the 
board,  and  a  the  angle  through  which  it  has  been  inclined,  then 
H/L  =  tan  a.  If  h  is  the  height  of  one  end  of  the  liquid  col- 
umn above  the  other  end,  and  1  is  the  slant  distance  between 
them,  h/1  =  sin  a.  Since  a  is  small,  we  take  the  sine  and  tan- 
gent as  equal,  hence  h  =  Hl/L  ;  and  since  the  pressure  equals 
Dhg,  where  D  is  the  density  of  the  liquid,  we  have 


i8]  VISCOSITY.    FLOW  OF  LIQUIDS  IN  TUBES.  33 

DgHl 
(2)         p  =    -£-. 

Ask  for  instructions  in  manipulating  both  pieces  of  apparatus. 

(a)  Blow  several  bubbles  of  different  sizes,  say  five  or  more, 
and  measure  the  pressure  and  diameter  of  each.  First  determine 
the  zero-readings  at  the  ends  of  the  liquid  column  in  the  pres- 
sure-gauge, then  blow  the  bubble,  put  it  in  communication  with 
the  gauge,  and  make  the  setting  as  indicated  above.  As  nearly 
at  the  same  time  as  possible,  measure  the  diameter  of  the  bub- 
ble by  making  settings  with  the  thread  on  each  side  of  the  bub- 
ble, putting  it  in  line  with  its  image  in  the  mirror  behind. 

(b)  Determine  whether  the  pressure  varies  inversely  as  the 
radius  of  the  bubble.  Calculate  for  each  case  the  surface  ten- 
sion of  the  soap  solution. 

18.     VISCOSITY.     FLOW  OF  LIQUIDS  IN  TUBES. 

References. — Watson,   p.    196;    Watson's   Practical   Physics,   p.    145; 
Ferry  and  Jones,   p.143. 

The  dependence,  of  the  rate  of  flow  in  tubes,  on  the  diameter 
and  length  of  the  tube,  and  on  the  temperature  of  the  liquid 
and  the  kind  of  liquid  used,  is  to  be  observed.  When  a 
liquid  flows  through  a  tube,  if  the  liquid  wets  the  walls  of 
the  tube,  the  layer  of  liquid  in  immediate  contact  with  the  wall 
generally  remains  at  rest.  The  speed  with  which  the  liquid 
moves  increases  from  the  surface  of  the  tube  to  the  axis  of  the 
tube.  Hence,  if.  we  imagine  the  liquid  to  consist  of  a  number 
of  hollow  cylinders  coaxial  with  the  tube,  the  fluid  within 
each  of  these  cylindrical  shells  will  be  moving  more  slowly 
than  in  the  shell  immediately  inside,  and  faster  than  in  the 
shell  immediately  outside.  This  relative  motion  of  adjacent 
layers  of  the  liquid  is  connected  with  the  internal  friction  or 
viscosity  of  the  liquid.  Viscosity  varies  greatly  with  the  kind 
of  liquid  used,  this  dependence  upon  the  character  of  the 
liquid  being  indicated  by  the  "coefficient  of  viscosity."  If  a 


34  VISCOSITY.      FLOW    OF   LIQUIDS   IN    TUBES.  fl8 

liquid  is  very  viscous,  like  syrup,  its  coefficient  of  viscositv  is 
high  ;  if  like  alcohol,  its  coefficient  of  viscosity  is  low.  For  a 
given  liquid  at  a  given  temperature,  the  coefficient  of  viscosity 
is  a  constant. 

In  the  case  of  a  liquid  flowing  through  a  long,  narrow  tube, 
the  volume  V,  issuing  per  second  from  the  end,  depends  upon 
the  difference  in  pressure  p,  between  the  two  ends  of  the  tube, 
the  radius  r  of  the  tube,  its  length  1,  and  the  coefficient  of  vis- 
cosity c  of  the  liquid.  These  quantities  are  connected  by  the 
relation 


V  == 

8lc* 

To  compare  the  coefficients  of  viscosity  of  two  different 
liquids,  it  is  evident,  if  the  above  relation  be  accepted,  that, 
for  equal  times  of  flowing,  the  coefficients  will  be  in  inverse 
proportion  to  the  volumes,  or  ct  :  c2  =  V2  :  Vj. 

Three  small-bore  tubes  are  provided,  two  being  of  the  same 
length  but  of  different  bores,  and  the  third  being  longer  but 
of  the  same  bore  as  one  of  the  two  shorter  ones.  The  reservoir 
used  consists  of  a  large  bottle  through  whose  cork  are  fitted 
two  glass  tubes,  long  enough  to  reach  about  two-thirds  of  the 
way  to  the  bottom.  The  outside  end  of  one  of  these  tubes 
is  connected  by  rubber  tubing  with  the  tube  through  which 
the  flow  is  to  be  measured  ;  the  other  tube  is  left  open  to  the 
air.  Both  tubes  must  extend  some  distance  below  the  level 
of  the  liquid  in  the  bottle,  and  the  cork  must  be  air-tight.  By 
means  of  this  arrangement  a  constant  head  of  pressure  may 
be  obtained. 

(a)  Clean  the  tubes  thoroughly  with  chromic  acid  and  rinse 
by  drawing  clean  water  through  them  with  a  jet-pump.  Attach 
one  of  the  tubes  to  the  siphon-tube  from  the  reservoir,  letting 
the  lower  end  dip  into  water  in  a  beaker.  Weigh  the  beaker 
and  contained  water  on  the  trip-scales.  Before  replacing  the 
beaker  in  position,  nearly  fill  the  reservoir  with  water  at  the 
room  temperature,  start  the  siphon,  and  let  the  water  run  into 


19]          RELATIVE  DENSITIES  OF  GASES.     TIME  OF  EFFLUX.  35 

a  waste  vessel  until  the  air  begins  to  bubble  from  the  lower  end 
of  the  open  tube  up  through  the  water  in  the  reservoir.  Then 
replace  the  beaker,  record  the  height  of  the  water-level  in  it, 
and  allow  the  water  to  flow  for  two  minutes.  Weigh  the  beaker 
again  to  determine  the  volume  which  has  run  through.  The 
head  of  pressure  will  be  given  by  the  difference  in  height  of 
the  lower  end  of  the  open  tube  in  the  reservoir  and  the  mean 
of  the  initial  and  final  levels  in  the  beaker.  Point  out  clearly 
why  the  head  is  measured  from  the  end  of  the  open  tube  and 
not  from  the  water-level  in  the  reservoir.  Make  two  indepen- 
dent trials. 

(b)  Repeat   with   each   of  the  other  tubes.      Measure   the 
diameters  of  the  tubes  with  the  micrometer  microscope,  or  by 
weighing  mercury  which  occupies  a  known  length  of  the  tube. 
What  do  your  results  show  concerning  the  dependence  of  the 
rate  of  flow  on  the  radius  and  length  of  the  tube?     Calculate 
the  coefficient  of  viscosity  of  the  water  for  the  three  cases, 
and  take  the  average  value. 

(c)  With  one  of  the  tubes,  use  water  at  5o°-6o°C.  in  the 
reservoir,  and  compare  with  previous  results  to  observe  the 
effect  of  temperature  on  viscosity. 

(d)  Repeat  (c)  with  a  ten  per  cent  solution  of  sugar,  and, 
if  there  is  time,  with  a  ten  per  cent  salt-solution.    Discuss  the 
results,  comparing  them  with  those  of   (a)   and   (b),  noting 
the  effect  upon  viscosity  of  different  sorts  of  dissolved  sub- 
stances. 

19.    RELATIVE  DENSITIES  OF  GASES.     TIME  OF 

EFFLUX. 

References. — Kohlrausch,  p.  64;  Ferry  and  Jones,  p.  107. 

The  ratio  of  the  densities  of  two  gases,  under  the  same 
conditions  as  to  pressure,  is  equal,  very  approximately,  to  the 
inverse  ratio  of  the  squares  of  the  speeds  with  which  the 


36        RELATIVE;  DENSITIES  OF  GASES.    TIME  OF  EFFLUX.        [19 

gases  escape  through  a  fine  opening  in  a  diaphragm.  Since 
the  time  of  escape  of  a  given  volume  will  be  inversely  as  the 
speed  of  efflux,  it  follows  that  the  ratio  of  the  densities  of  two 
gases  is  equal  to  the  direct  ratio  of  the  squares  of  the  time  of 
efHux  of  equal  volumes  under  the  same  conditions.  This  rela- 
tion was  experimentally  discovered  by  Bunsen.  For  a  proof 
of  it,  from  the  energy  relations,  see  the  second  reference  given 
above. 

(a)  The  gas-holder  consists  of  a  glass  cylinder,  at  the  top 
of  which  is  a  three-way  stop-cock  and  a  diaphragm  with  a 
fine  opening.     The  cylinder  is  placed  in  a  reservoir  of  mer- 
cury.    The  three-way  cock  allows  communication  to  be  made 
with  the  outside  for  filling  or  with  the  diaphragm.  Within  the 
cylinder  is  a  float  which  indicates  when  the  desired  volume  of 
gas  has  escaped. 

First  fill  the  cylinder  with  dry  air.  To  do  this,  turn  the 
stop-cock  so  as  to  put  the  cylinder  in  communication  with  the 
air,  and  lower  the  cylinder  as  far  as  it  will  go.  This  drives 
out  most  of  the  contained  gas.  Connect  the  cylinder  with  a 
calcium  chloride  drying-tube,  and  raise  the  cylinder.  This 
operation  will  fill  the  cylinder,  and  by  repeatedly  emptying 
and  filling  the  cylinder  it  will  become  practically  freed  of  the 
moist  air  or  other  gas  previously  contained  in  it.  Close  the 
stop-cock,  and  lowering  the  cylinder,  clamp  it  in  position. 
Turning  the  stop-cock  so  that  the  gas  in  the  cylinder  is  in  com- 
munication with  the  diaphragm,  note  the  time  when  the  upper 
point  of  the  float  is  on  a  level  with  the  surface  of  the  mercury 
or  with  a  mark  on  the  cylinder.  Again  note  the  time  when  the 
second  mark  on  the  float  is  on  the  same  level.  Repeat,  mak- 
ing two  or  three  determinations  of  the  time  of  efflux  for  the 
given  volume  of  air,  and  take  the  mean. 

(b)  Repeat   (a),  filling  the  cylinder  with  illuminating  gas, 
following  the  directions  there  given   for  filling  the  cylinder, 
the  cylinder  being  connected  directly  to  the  source  of  the  gas 
used.     Note  the  time  of  efflux  between  the  same  two  positions 


2OJ  CALIBRATION   OF  A   THERMOMETER,  ABSOLUTE.  37 

for  the  float  as  used  in  (a).    This  insures  the  same  conditions 
as  to  pressure  in  the  two  cases. 

(c)  Repeat   (b),  using  dry  carbon  dioxide. 

(d)  Calculate  the  relative  densities,  referred  to  air,  of  the 
gases  used  in    (b)    and   (c).     Taking  the  density  of  dry  air 
under  standard  conditions  to  be  0.001293  gms.  per  cc.,  find 
the  density,    under    standard    conditions,  of   the    gases    used. 
What   "Laws"   have   been   used,   or     assumptions     made,   in 
answering  the  requirement  of  the  preceding  sentence? 

20.     CALIBRATION  OF  A  THERMOMETER, 
ABSOLUTE. 

References. — Kohlrausch,    p.    81    and    p.     86;     Watson's    Practical 
Physics,  p.   162;  Miller,  p.   160;   Edser,  p.  23. 

The  object  is  to  determine  the  fixed  points  and  the  correc- 
tions to  the  scale-readings  of  a  mercurial  thermometer. 

(a)  Determination    of   the   Lower   Fixed   Point. — Put   the 
thermometer  through  the  cork  in  a  test-tube,  having  filled  the 
latter  about  half  full  of  distilled  water.     Place  the  tube  in  a 
freezing  mixture  of  shaved  ice  and  salt,  and  stir  the  water 
around  the  thermometer  until  it  begins  to  freeze.     Read  the 
thermometer.    By  warming  the  tube  in  the  hand  and  repeating 
the  freezing  process,  obtain  several  readings  of  the  true  zero- 
point. 

(b)  Determination   of  the   Upper  Fixed  Point. — Place  the 
thermometer  in  the  cork  in  the  tube  at  the  top  of  the  boiler, 
though  with  the  bulb  well  above  the  surface  of  the  water. 
Boil  the  water  so  that  the  steam  issues  freely,  but  not  with 
any  perceptible    pressure,    from    the    upper    vent.     Read    the 
thermometer  when   it  becomes   steady.     Allow   the  boiler  to 
cool  slightly  and  repeat,  making  in  all  three  readings.     If  the 
instrument  be  provided  with  a  water-manometer,     take     the 
manometer  reading  simultaneously  with  the  temperature  read- 
ing.    Read  the  barometer  and  determine  the  pressure  of  the 


3^  CAU1JKATION   OF  A  THERMOMETER,   ABSOLUTE.  [2O 

steam,  and  find  the  true  boiling1  point  for  this  pressure  from 
the  Tables. 

(c)  Let  the  thermometer  cool  slowly  to  about  the  tempera- 
ture of  the  room,  and  repeat  (a).     If  the  freezing  point  ob- 
served now  is  different   from  that  observed  in   (a),  use  the 
mean  of  the  two  values  in  the  calibration  that  follows.  Assum- 
ing the  correct  freezing  point  to  be  o°,  write  the  corrections 
of  the  thermometer  at  the  zero-point  and  at  the  boiling  point 
Record  these  two  corrections  by  points  on  coordinate  paper, 
having  as  abscissae  degrees  centigrade  from  o°  to  110°,  and  as 
ordinates   the   corrections   of  the  thermometer   at   the   corre- 
sponding temperatures  in  tenths  of  a  degree  but  on  a  larger 
scale.    Corrections  should  be  plus  (  +  )  if  they  are  to  be  added 
to  the  observed  to  give  the  true  temperatures,  minus   ( — )   if 
they  are  to  be  subtracted.     Connect  these  two  points   by   a 
straight  line.     The  ordinate  of  this  straight  line  at  any  point 
gives  the  correction  of  the  thermometer  at  that  temperature 
on  the  assumption  that  the  bore  of  the  thermometer  is  uni- 
form throughout  the  whole  range.     In  general  this  assump- 
tion is  not  justified,  and  there  must  be  added  to  this  correc- 
tion at  each  point  another  correction  due  to  the  inequalities  of 
the  diametef  of  the  bore.     In  order  to     determine    this  latter 
correction  it  will  be  necessary  to  calibrate  the  tube. 

(d)  Calibration  of  the  Tube. — Break  off  a  portion  of  the 
thread  of  mercury  about  ten  degrees  in  length.     ( Both  Kohl- 
rausch  and  Miller  give  directions  for  breaking  the  thread  at 
any  desired  point,  but  if  you  cannot  readily  succeed,  ask  for 
assistance.)     Place  the  lower  end  of  the  thread,  approximately 
ten  degrees  long,  at  the  zero-point  of  the  scale  and  read  the 
position  of  the  upper  end  to  tenths  of  a  degree.     Then  place 
the  lower  end  at  10°  and  read  the  upper  end.    Repeat  with  the 
lower  end  at  the  successive  points  20°,  30°,  40°,  etc.,  up  to 
90°  ;  then  come  down  again  with  upper  end  at  100°,  90°,  80°, 
etc.,  reading  the  lower  end  each  time. 


2O]  CALIBRATION   OF   A   T  H^RMOMKTER,   ABSOLUTE.  39 

.   (c)   Record    the    observations    and    calculations    in    tabular 
form  in  six  columns  as  follows : 

1 i )  The  reading  of  the  lower  end,  o,  10,  20,  30,  etc. 

(2)  The  reading  of  the  upper  end. 

(3)  The  length  of  the  thread  in  each  position. 

(4)  The  mean  length,  1,  for  each  position. 

By  the  mean  length,  1,  for  each  position  is  meant  the  mean 
of  the  readings  over  a  certain  interval  going  up  (say  from 
30  to  40)  and  over  the  same  interval  (40  to  30)  coming 
down.  Find  the  mean  value  of  all  these  lengths  through- 
out the  whole  range  and  record  this  as  the  "mean  length,"  L. 

(5)  The  correction,   (L  —  1),  for  the  length  of  each  inter- 
val, that  is,  the  difference  between  the  mean  length  for  all 
intervals  and  the  observed  length  of  each  interval. 

(6)  The  correction  for  the  upper  end  of  each  interval.   This 
is  the  correction  for  the  lower  end  of  the  interval  plus  the 
correction  for  the  length  of  the  interval,  since  a  correction  at 
any  point  evidently  affects   all   points  above  this.     The  cor- 
rection  thus    found   for  any   point   represents   the  magnitude 
of  the  inequalities  of  the  bore  up  to  that  point.     It  must  be 
added   to   the   observed    reading   for   that    point   to   give   the 
correct   reading,   it  being  assumed   that  the  fixed  points   are 
properly   placed.      The   corrections   should   be   recorded   with 
proper  signs.     (See  Watson's  Practical  Physics,  p.  168.) 

(/)  To  construct  a  final  table  of  corrections  it  is  neces- 
sary to  add,  algebraically,  the  corrections  found  in  (c)  and 
in  (c,  6).  This  can  best  be  done  by  plotting.  On  the  plot 
made  in  (c)  record  at  the  points  10°,  20°,  etc.,  the  correc- 
tions at  these  points  obtained  from  (e),  measuring  the  cor- 
rections up  or  down  from  the  slanting  line  already  drawn, 
according  as  the  sign  is  plus  or  minus.  Draw  a  smooth 
curve  through  the  points.  The  ordinate  of  this  curve  at  any 
point,  measured  from  the  horizontal  base  line,  is  the  total 
correction  to  the  reading  of  the  thermometer  at  the  corres- 
ponding temperature. 


40  CALIBRATION  OF  A  THERMOMETER,  RELATIVE.  [21 

21.     CALIBRATION  OF  A  THERMOMETER, 
RELATIVE. 

Most  varieties  of  glass  expand  at  different  rates  at  differ- 
ent temperatures,  hence,  even  with  a  thermometer  whose  bore 
has  been  carefully  calibrated  by  some  such  method  as  given  in 
Exp.  20,  the  reading  can  be  relied  upon  only  within  certain 
limits.  After  having  obtained  a  thermometer  whose  calibra- 
tion curve  is  accurately  known,  so  that  it  may  be  taken  as  a 
"standard,"  the  most  convenient  method  of  calibrating  other 
thermometers  is  by  direct  comparison  with  the  standard.  If 
the  calibration  curve  of  the  standard  thermometer  can  be 
relied  upon,  all  irregularities  of  the  thermometer  calibrated 
can  be  corrected. 

The  thermometer  to  be  calibrated  in  this  experiment  is  a 
50°  thermometer  reading  to  o°.i.  Tie  the  thermometer  to 
be  calibrated  to  the  "standard"  with  soft  cotton  twine,  wind- 
ing it  between  the  stems  so  as  to  separate  them  slightly.  Put 
the  bulbs  nearly  opposite  each  other ;  and  see  that  correspond- 
ing divisions  are  as  nearly  opposite  as  is  consistent  with  this 
condition.  Suspend  the  two  securely,  with  the  bulbs  in  the 
middle  of  a  kettle  of  water,  and  steady  the  stems  by  catching 
them  loosely,  without  pressure,  in  a  clamp.  The  thermom- 
eters are  to  be  read  by  a  short-focus  telescope,  which  slides 
easily  on  the  vertical  rod  of  its  stand.  This  should  be  set 
with  its  object-glass  at  a  distance  of  about  50  cm.  from  the 
thermometers,  which  should  be  perpendicular  to  its  axis. 
When  taking  a  reading  always  set  the  telescope  so  that  the 
top  of  the  mercury  column  appears  in  the  middle  of  the  field 
of  view,  not  near  its  upper  or  lower  edge,  in  order  to  avoid 
parallax. 

(a)  Take  a  careful  series  of  readings  to  o°.oi.  at  intervals 
of  2°  or  3°  from  about  5°  to  45°.  Keep  the  water  well  stirred, 
and  keep  the  temperature  fairly  constant  for  a  few  minutes 
before  each  reading.  A  good  plan  is  to  take  a  preliminary 


22]  VARIATION  OF  BOILING  POINT  WITH   PRESSURE.  41 

reading  of  each  thermometer  in  order  to  see  about  where  the 
reading  is  going  to  come.  The  two  exact  readings  can  then 
be  made  so  quickly  as  to  be  practically  simultaneous.  Read 
again  in  a  few  seconds,  taking  the  thermometers  in  reverse 
order.  Repeat  if  necessary  until  the  differences  obtained  for 
two  such  readings  agree  fairly  well. 

(b)  Let  the  observers   change  places,   and  take  a  similar 
descending  series,  cooling  the  water  by  dipping  out  hot  and 
adding  cold  water. 

(c)  Ask  to  see  the  calibration  curve  of  the  standard  used, 
and  from  it  construct  a  table  of  corrections  for  the  thermom- 
eter you  are  calibrating.  Plot  a  calibration  curve,  recording  the 
number  of  the  thermometer.    In  your  future  work  with  a  ther- 
mometer of  this  type,  use  the  one  you  have  calibrated. 

22.     VARIATION  OF  BOILING  POINT  WITH 
PRESSURE. 

References. — Edser,  p.   191;  Millikan,  p.  157. 

If,  after  reading  the  references,  the  arrangement  of  the 
apparatus  is  not  understood,  ask  an  assistant  for  directions. 
Before  turning  off  the  air-pump,  be  sure  to  let  air  into  the  ap- 
paratus by  opening  the  pinch-cock  nearest  the  pump,  other- 
wise water  will  flow  back  into  the  tubing.  In  boiling  the 
water  do  not  play  the  flame  on  the  flask  directly  below  the 
glass  beads,  but  rather  to  the  side,  but  never  above  the  water- 
line. 

(a)  The  water  should  first  be  started  through  the  steam 
condenser.  This  is  a  glass  or  metal  tube  used  to  jacket  the 
tube  leading  from  the  boiling-flask,  thus  condensing  the  steam 
as  it  comes  from  the  flask.  The  thermometer  should  be  passed 
through  the  stopper  of  the  flask  and  so  regulated  that  its  bulb 
will  be  in  the  rising  steam,  but  not  in  the  water.  The  con- 
nection with  the  large  glass  bottle  serves  to  equalize  sudden 
changes  in  pressure. 


42      EXPANSION  OK  A  i.inrin  r,v  ARCH IMKDKS'  PKI  \an.K.       [23 

First  boil  the  water  at  atmospheric  pressure,  reading  the 
manometer  and  noting  the  temperature.  Then  take  a  series 
of  readings  at  intervals  of  about  5  cm.  pressure,  until  the 
"bumping"  becomes  so  violent  as  to  render  further  reading 
impracticable.  Before  each  reading,  after  pumping  to  the  pres- 
sure desired,  close  the  stop-cock  over  the  jet-pump,  wait  a 
short  time  for  the  pressure  to  reach  equilibrium,  and  then 
make  the  reading  of  boiler  temperature  and  corresponding 
pressure.  Put  the  pump  again  in  connection,  obtain  a  new 
pressure,  and  repeat  the  readings. 

(b)  Take   a  series   of   readings   with   increasing   pressures 
up  to  atmospheric  pressure,  choosing  values  different  from  the 
previous  ones. 

(c)  Plot  the  observations  on  coordinate  paper,  using  pres- 
sures as  ordinates  and  temperatures  as  abscissae.     From  the 
curve  find  the  boiling  point  of  water  at  a  pressure  of   1/2 
atmosphere.  Discuss  the  phenomena  of  this  experiment  in  con- 
nection  with  the   difficulties   experienced   in   cooking   food   at 
high  altitudes.     Could  determinations  of  the  boiling  point  of 
water  be  used  to  measure  altitude,  and  how  ?  . 

23.    COEFFICIENT  OF  EXPANSION   OF  A  LIQUID 
BY   ARCHIMEDES'   PRINCIPLE. 

The  coefficient  of  expansion  of  a  heavy  oil  is  to  be  obtained 
by  observing  the  change  in  the  buoyant  force  acting  on  a  metal 
cylinder  when  immersed  in  the  oil  at  different  temperatures. 
A  brass  cylinder  is  suspended  from  one  arm  of  the  balance 
and  carefully  weighed,  first  in  air,  then  in  water  at  a  known 
temperature.  The  oil  is  then  placed  in  a  calorimeter  con- 
sisting of  one  beaker  inside  another,  and  the  cylinder  is 
weighed  when  immersed  in  the  oil,  the  temperature  of  the  oil 
being  noted,  which  should  be  the  same  as  that  of  the  water, 
or  nearly  so.  Since  the  oil  thickens  if  cooled,  it  is  convenient 
to  make  the  first  weighings  at  the  room  temperature. 


24]     COMPARISON  OF  ALCOHOL-  AND  WATER-THERMOMETERS.    43 

After  weighing  in  the  cool  oil,  the  inner  beaker  is  removed 
and  placed  in  a  water-bath  heated  to  60°  or  70° C.  Replacing 
the  beaker  with  the  heated  oil  in  the  calorimeter  beaker,  the 
cylinder  is  again  weighed  in  the  oil,  the  temperature  of  the  oil 
during  the  weighing  being  carefully  noted. 

Let  M  =  the  mass  balancing  the  cylinder  when  in  air, 
mt  =  the  mass  balancing  the  cylinder  when  in  water, 
nio  =  the  mass  balancing  the  cylinder  when  in  cool  oil, 
m.,  =  the  mass  balancing  the  cylinder  when  in  hot  oil, 
t1  =  the  temperature  of  the  cool  oil  and  the  water, 

and    t2  =  the  temperature  of  the  hot  oil. 

Then  Vlf  the  volume  of  the  cylinder  at  the  lower  temperature, 
is  (M  —  mJ/D,  where  D  is  the  density  of  the  water  at  this 
temperature  (see  Tables).  If  V2  is  the  volume  of  the  cylinder 
at  the  higher  temperature,  t,,  and  a  the  coefficient  of  cubical 
expansion  (3  times  the  coefficient  of  linear  expansion  found 
in  the  Tables),  we  have  V2  =  V,  f  I  +  a(t,  —  tt)].  The  den- 
sity, D,,  of  the  cool  oil  is  (M  —  m2)/V1.  The  density,  D2, 
of  the  hot  oil  is  (M  —  m,,)/V2.  If  ft  be  the  coefficient  of 
cubical  expansion  of  the  oil  for  the  temperature  range  used, 
we  have  D1  =  DJ I  -f-  p(t.2  —  t,)],  from  which  (3  can  be 
found. 

Make  two  determinations  of  the  density  of  the  oil  at  the  two 
temperatures.  From  the  mean  of  the  values  for  each  density 
calculate  ft. 

Point  out  the  sources  of  error  in  the  experiment.  If  the 
cylinder  had  an  internal  cavity,  show  what  its  effect  upon  the 
value  of  /?  would  be. 

24.     COMPARISON  OF  ALCOHOL-  AND  WATER- 
THERMOMETERS. 

In  this  experiment  the  relative  expansions  of  water  and 
alcohol  are  to  be  studied,  and  the  behavior  of  these  liquids 
when  used  in  thermometers  to  be  observed. 


44     COMPARISON  OF  ALCOHOL- AND  \\ATER-TH ERMOMETERS.     [24 

(a)  Two  thermometer  bulbs  are  to  be  filled,  one  with  water, 
the  other  with  ethyl  alcohol  by  the  aid  of  the  reservoir-tube. 
The  reservoir  is  fitted  on  the  end  of  the  thermometer-stem,  filled 
with  water  (or  alcohol),  and  warmed.  The  liquid  is  then 
introduced  into  the  thermometer-bulb  by  alternately  heating 
the  bulb  to  drive  out  the  air  and  allowing  it  to  cool  to  admit 
the  liquid.  Ask  for  assistance  if  the'  method  is  not  understood. 
The  water  should  be  heated  to  drive  out  the  oxygen  held  in  so- 
lution, before  filling  the  reservoir  with  it.  Take  care  not  to 
ignite  the  alcohol.  The  liquid  in  each  thermometer  should 
stand  i  or  2  cm.  above  the  lower  end  of  the  stern  when  the 
bulb  is  in  melting  ice. 

(b)  Glue  or  otherwise  fasten  a  strip  of  stiff  paper  along  the 
back  of  each  stem,  to  be  used  as  a  scale.    Then  place  the  ther- 
mometers in  clamps  with  their  bulbs  in  shaved  ice  or  in  a  mix- 
ture of  water  and  ice.    When  the  reading  becomes  steady,  indi- 
cate the  position  of  the  meniscus  of  each  by  a  sharp  line  on  the 
card.     Mark  the  line  zero.    This  is  the  first  fixed  point  of  the 
thermometer. 

(c)  To  determine  the  second  fixed  point,  place  the  bulbs 
in  a  beaker  of  wood  alcohol  which  is  itself  placed  on  a  sup- 
port in  a  bath  of  water.     Heat  the  water-bath  slowly  until  the 
wood  alcohol  begins  to  boil.     Be  very  careful  not  to  bring  the 
alcohol  itself  to  the  flame,  and  avoid  inhaling  the  fumes  of 
wood   alcohol.     When   steady,  again   indicate  the  position  of 
the  meniscus  on  each  stem  by  a  sharp  line.     Mark  this  point 
66,  which  is  the  boiling  point  of  wood  alcohol  on  the  centi- 
grade scale. 

(d)  Lay  the  stems  of  the  thermometers  on  a  flat  surface, 
measure  the  distance  on  each  between  the  two  fixed  points, 
and  divide  this  distance  into  66  equal  parts,  calling  each  part 
a  degree.     Put  the  marks  for  each  degree  on  the  scale  and 
number  every  tenth  one. 

(e)  Place   the  two   arbitrarily   calibrated   thermometers   in 
a  water  bath  of  o°,  as  recorded  by  each  thermometer.     Gradu- 


25]  COEFFICIENT  OF  EXPANSION  OF  MERCURY.  45 

ally  raise  the  temperature  of  the  water-bath  and  note  the  read- 
ings of  the  two  thermometers,  at  first  at  short  intervals,  then 
at  longer  intervals,  until-  the  upper  fixed  point  is  reached. 

(/)  Plot  a  curve  on  coordinate  paper,  having  as  abscissae 
the  temperatures  by  the  alcohol  thermometer,  and  as 
ordinates  the  corresponding  temperatures  by  the  water 
thermometer.  Draw  a  smooth  curve  through  the 
points.  This  curve  gives  the  relation  between  the  tem- 
peratures as  recorded  by  the  two  thermometers.  What 
inferences  can  you  draw  from  the  curve?  If  alcohol  be 
taken  as  the  standard  substance,  what  can  you  say  of  the  uni- 
formity of  the  expansion  of  the  water?  If  the  water  be 
assumed  as  the  standard,  what  of  the  expansion  of  the  alco- 
hol ?  Which  would  be  the  best  substance  to  use  in  a  practical 
thermometer,  and  why? 

25.    COEFFICIENT  OF  EXPANSION  OF  MERCURY 

BY  REGNAULT'S  METHOD. 
References. — Watson,   p.   221;   Edser,  p.  71. 

This  method  was  originally  devised  by  Dulong  and  Petit, 
but  improved  and  made  practical  by  Regnault.  It  is  an  abso- 
lute method  in  which  the  effect  of  the  expansion  of  the  con- 
taining vessel  is  eliminated.  The  method  is  applicable  to  any 
liquid.  Two  glass  tubes,  AA'  and  BB',  are  surrounded  by 
metal  cylinders,  L  and  M  respectively,  in  which  baths  of  dif- 
ferent temperatures  may  be  placed.  These  glass  tubes  are 
connected  by  a  horizontal  tube,  ACB,  (from  which  there  ex- 
tends an  upright  open  tube,  C),  and  also  by  an  inverted  U-tube 
at  the  bottom  between  A'  and  B'.  Mercury  is  poured  into  the 
glass  tubes  until  it  stands  at  some  point  in  C  just  above  the 
horizontal  level,  AB.  This  insures  the  height  remaining  the 
same,  or  very  nearly  the  same,  at  A  and  B,  and  gives  a  means 
of  measuring  the  height  without  observing  the  meniscus  at 
A  or  B.  In  the  bend  of  the  tube  GK  there  is  compressed  air 


46 


COEFFICIENT  OF  EXPANSION  OF  MERCURY. 


[25 


so  that  the  pressure  is  always  the  same  at  the  meniscus  G  and 
the  meniscus  K.  The  levels  G,  K,  and  C  may  be  measured. 
Cold  water  is  passed  through  M,  or  a  mixture  of  ice  and 
water  placed  in  it,  and  steam  is  passed  through  L,  thermome- 
ters at  A  and  B  indicating  the  temperatures.  Suppose  that 


the  temperature  of  M  is  o°C.  and  that  of  L,  t^0.  The  temper- 
ature of  the  mercury  in  both  branches  of  the  inverted  U-tube 
may  be  assumed  to  be  the  same  as  that  of  M.  Let  D0  be  the 
density  of  the  mercury  in  M  (at  o°C.),  and  Dl  be  the  density 
of  the  mercury  in  L  (at  t^C.),  H  the  vertical  height  of  C 
above  the  level  of  A'B',  and  h  the  vertical  distance  OK.  Then, 
since  the  pressure  at  G  is  the  same  as  that  at  K,  we  have  in 
the  tube  A'E  a  pressure  one  way  due  to  the  hot  mercury  col- 
umn AA',  and  balancing  it  a  pressure  due  to  the  cold  column 
equivalent  to  (BB'  —  DK  +  GE),  that  is,  to  a  column  of 
height  (BB'  — GK)  or  (H  — h).  Therefore,  if  P0  be  the 
atmospheric  pressure,  we  have, 

D,SH  +  P0  =  D0g(H  —  h)  +  P«. 
from  which 

(i)         D0/D1  =  H/(H  —  h). 

Now,  if  ft  be  the  coefficient  of  cubical  expansion  of  mercury, 
v,,  the  volume  of  a  given  mass  m  at  o°C.,  v,  the  volume  of  the 


25]  COEFFICIENT  OF  EXPANSION  OF  MERCURY.  47 

same  mass  at  t,°C.,  we  have  vl  =  v,,(i  -\-ftti).  Since  v0  = 
m/D0  and  v,  =111/0,,  Z^*-:pk{*-Hh  ftt,)  or  i  +  0tt  =  D0/D1 
=H/(H  —  h). 

Therefore     ( 2 )          /?  =h/  ( H  —  h  )  tt . 

If  the  mercury  in  M  and  in  the  U-tube  is  at  room  tempera- 
ture (.say  to0)  instead  of  at  o°,  ft  calculated  from  (2)  will  be 
the  average  coefficient  of  cubical  expansion  between  t,°  and  tl°, 
and  not  the  coefficient  between  o°  and  t^  (or  100°)  as  given 
in  the  Tables.  In  order  to  find  this  zero  coefficient  a  slightly 
different  equation  should  be  used.  We  not  only  have 
D0  =  D,(i  +  jSti)  and  D0  =  D2(i  +  0t2),  but  also  D2/Dl  = 
H/(H  —  h),  where  D2  is  the  density  at  t2°.  From  these  three 
relations  (3  is  obtained. 

(a)  Pass   cold   running  water  through   the  jacket   M   and 
steam  through    L,   keeping  thermometers   on    the   two   sides. 
Wrap  a  cloth  about  the  horizontal  tube  A'E,  and  keep  this 
wet  with  cold  water  to  make  the  conduction  of  heat  to  the 
column  EG  as  small  as  possible.     Note  the  difference  in  height 
of  the  menisci   at  G  and   K  when  the  temperatures  become 
steady.     The  distance,  h,  must  be  very  accurately  determined 
at  each  setting,  whereas  a  reading  of  the  height,  H,  to  I  mm. 
is  sufficiently  accurate.     Do  not  forget  to  read  both  heights 
and  the  temperatures  when  there  is  a  change. 

(b)  The  pressure  of  the  gas  in  the  inverted  U-tube,  GK, 
should  now  be  changed.  Do  not  attempt  to  do  this  yourself  until 
an  assistant  has  shown  you  the  method.     The  height  of  the 
mercury  in  C  may  require  adjusting  by  adding  or  taking  out 
a  little  mercury.     Only  perfectly  clean  mercury  should  ever 
be  used.  Read  h  and  H  again,  making  four  or  five  different  set- 
tings. 

(d)  Calculate  for  each  setting  the  value  of  the  mean  coeffi- 
cient of  expansion  between  the  temperature  of  cold  water  and 
the  steam,  and  take  the  mean. 

Derive  the  equation  for  ft,  the  average  coefficient  of  cubical 


48  EXPANSION  OF  GLASS  BY  WEIGHT  THERMOMETER.  |>6 

expansion  between  o°C.  and  ioo°C.,  and  calculate  its  value 
from  your  results.  What  percentage  difference  is  there  between 
this  value  of  ft  and  the  one  calculated  by  equation  (2)  ? 

Calculate  the  percentage  error  in  your  value  of  ft,  taking 
the  value  from  the  Smithsonian  Tables  as  the  correct  one. 

Considering  AA'EG  as  one  U-tube  with  balancing  columns 
of  liquid,  write  the  equation  for  equal  pressures  at  the  level 
A'E ;  do  the  same  for  the  branch  BB'DK,  and  from  these  see 
if  equation  (i)  can  be  derived. 

26.     EXPANSION  OF  GLASS  BY  WEIGHT 
THERMOMETER. 

References. — Watson,   p,  219;   Watson's   Practical   Physics,  p.   195; 

Edser,  p.  66. 

The  purpose  of  this  experiment  is  to  determine  the  coeffi- 
cient of  cubical  expansion  of  glass  by  means  of  the  weight 
thermometer. 

The  weight  thermometer  consists  of  a  glass  tube  closed  at 
one  end  and  ending  in  a  curved  capillary  at  the  other  end.  It 
is  filled  with  mercury  at  o°C.,  and  the  mass  of  the  mercury 
measured.  When  later  placed  in  a  bath  of  higher  tempera- 
ture, some  mercury  overflows,  since  mercury  expands  more 
rapidly  when  heated  than  does  glass.  The  mass  of  this  over- 
flow is  measured. 

Let  M  =  the  mass  of  the  mercury  filling  the  weight  ther- 
mometer at  o°C., 

V0  =  the  volume  of  M,  and  hence  of  the  weight  thermom- 
eter at  o°C, 

^  and  y  =  the  coefficients  of  expansion  respectively  of  mer- 
cury and  glass, 

and  m=    the  mass  of  the  mercury  which  overflows  when 
the  temperature  is  raised  from  o°  to  t°. 
Then        V0(i  +  ftt)  =  the  volume  of  the  mass    M   at  t°. 


26]  EXPANSION  OE  GLASS  BY  WEIGHT  THERMOMETER.  49 

and  V0(  i  -f-  yt)  =  the  volume  of  the  weight  thermometer  at  t°  ; 
hence  V0(i  +  0t)  —  V0(  I  +  yt)  ==  V0(/3  —  y)t  =  the  vol- 
ume of  the  mass  m  at  t°.  If  D0  and  Dt  be  the  densities  of 
mercury  at  o°  and  t°C.,  respectively,  then 

(1)  D0  =  M/V0, 

(2)  Dt  =  m/V0(0  —  y)t, 

(3)  D0  =  Dt(i+0t). 

By  eliminating-  D0  and  Dt  from  equations  (i),  (2),  and  (3), 
we  get 


Besides  containing  the  known  masses,  M  and  m,  this  equation 
contains  the  three  quantities,  /?,  y,  and  t.  Any  two  of  these 
three  quantities  being  known,  the  third  will  be  given  by  the 
equation.  In  the  present  experiment  ft  and  t  are  known,  and 
y  is  to  be  calculated. 

(a)  Weigh  the  empty  weight  thermometer  to  10  mg.  Then  fill 
it  with  mercury.  In  doing  so  it  should  be  held  by  a  clamp,  or 
suspended  in  a  gauze  jacket,  and  heated  by  a  flame  held  in  the 
hand.     The  end  of  the  capillary   dips   under  the   surface  of 
mercury  in  a  porcelain  dish.     The  mercury  in  this  dish  should 
first  be  heated,  and  then  the  weight  thermometer  heated  until 
the  air  bubbles  out  through  the  mercury.     On  allowing  the 
bulb  to  cool,  some  mercury  will  run  into  it.     The  process  is 
then  repeated.     When  considerable  mercury   is  in   the  bulb, 
heat  it  until  it  boils  vigorously,   but  be  careful   not  to  heat 
too  hot  that  portion  of  the  glass  where  there  is  no  mercury. 
Keep  the  mercury  in  the  dish  hot,  otherwise  the  glass  is  apt 
to  crack  just  as  the  cooler  mercury  rushes  in.     The  tube  must 
be  completely  filled  to  the  end  of  the  capillary,  the  last  bubble 
of  air  being  expelled. 

(b)  Keeping  the  end  of  the  capillary  in  the  dish  of  mer- 
cury, allow  the  weight  thermometer  to  cool  in  the  air  suffi- 
ciently so  that  you  can  bear  your  hand  on  it.     Then  surround 


5O         EXPANSION    OF   A   LIQUID   HY    PYCNOMETER    MKTHOD.       [2/ 

it  with  shaved  ice  and  leave  it  long-  enough  to 
contract  as  much  as  it  will.  Assume  that  its  tem- 
perature is  now  o°  C.  Carefully  remove  the  dish 
and  brush  the  mercury  off  the  end  of  the  capillary. 
Place  a  watch-glass  under  the  end  to  catch  the  mercury  as  it 
begins  to  expand  and  flow  out.  Now  remove  the  ice-bath  and 
warm  the  bulb  with  the  hand  until  its  temperature  is  raised 
to  the  temperature  of  the  room. 

(c)  Place  the  weight  thermometer  in  the  boiler  provided.  Heat 
it  to  the  boiling  point  of  water  by  passing  steam  over  it  until  no 
more  mercury  comes  out.  Read  the  barometer  and  calculate 
the  temperature  of  the  steam.  Very  carefully  weigh  the  mer- 
cury in  the  watch-glass  to  i  mg.  Weigh  the  weight  ther- 
mometer and  contained  mercury  to  10  mg. 

(d)  Using  your  values  of  M  and  m,  and  taking  the  coeffi- 
cient of  expansion  of  mercury  as  found  in  Exp.  25,  calculate 
the  coefficient  of  cubical  expansion  of  glass. 

What  additional  measurements  would  you  need  to  make  in 
order  to  measure  the  room  temperature  with  your  weight 
thermometer  ? 

27.     EXPANSION  OF  A  LIQUID  BY  PYCNOMETER 

METHOD. 

Reference. — Edser,  p.  81  and  p.  86. 

The  method  consists  in  determining  the  mass  of  the  liquid 
(alcohol)  filling  a  pycnometer  at  each  of  several  different  tem- 
peratures, and  from  the  data  calculating  the  coefficient  of  ex- 
pansion of  the  liquid.  Four  determinations  should  be  made, 
at  intervals  of  about  8°.  beginning  with  the  room  tempera- 
ture. 

(a)  Fill  the  pycnometer  with  alcohol  and  set  it  on  a  plat- 
form in  a  kettle  of  water,  so  that  the  water  comes  well  up  to 
the  neck  of  the  pycnometer.  Hang  a  50°  thermometer  in  the 
bath  alongside  the  pycnometer,  and  keep  the  bath  well  stirred 


27]  EXPANSION  OF  A  LIQUID  BY  PYCNOMETER  METHOD.         51 

for  about  five  minutes.  The  temperature  of  the  bath,  which 
must  be  a  little  above  that  of  the  room,  should  remain  con- 
stant within  o°.i  during  this  time,  and  at  the  end  of  it  the 
alcohol  will  have  the  same  temperature  within  o°.i.  Take 
the  pycnometer  out  of  the  bath,  wipe  the  outside  dry,  and 
weigh  to  i  mg. 

(b)  Repeat  with  the  bath  at  about  each  of  the  higher  tem- 
peratures  selected,    holding  the   temperature   steady    for   ten 
minutes  by  holding  the  lamp  under  the  kettle  for  a  few  sec- 
onds  occasionally.     Careful  trial   has  shown  that    after  this 
treatment   the  temperature  of  the  alcohol  at  the  center  of  the 
pycnometer  is  about  o°.i   lower  than  that  of  the  bath,   and 
therefore  the  average  temperature  of  the  alcohol  is  the  same 
as  that  of  the  bath  to  within  o°.i. 

(c)  Empty  the  alcohol  into  the  bottle  from  which  it  was 
taken,  dry  the  pycnometer  with  a  jet-pump,  and  weigh. 

(d)  Determine  the  mass  of  the  alcohol  filling  the  pycnom- 
eter  at   each   temperature.      Plot   the   results,   with   tempera- 
tures, starting  from  o°,  as  abscissae,  and  changes  in  mass  as 
ordinates.      Assuming  that  the   expansion   is   uniform,   draw 
the  straight  line  which  best  represents  the  plotted  points,  and 
from  it  find  the  mass  filling  the  pycnometer  at  o°. 

(e)  From  the  masses  filling  the  pycnometer  at  o°  and  at 
40°,  calculate  the  relative  volume  of  a  given  mass  of  alcohol 
at  40°,  refered  to  its  volume  at  o°.     From  this  calculate  the 
coefficient  of  cubical  expansion  of  the  alcohol  for  the  given 
range  of  temperature. 

(/)  What  has  been  found  is  not  the  absolute  coefficient  of 
expansion,  since  the  pycnometer  also  expands.  Find,  from 
your  own  work  in  Exp.  26,  or  from  the  Tables,  the  coefficient 
of  cubical  expansion  of  glass,  and  by  applying  it  to  the  above 
result  find  the  absolute  coefficient  for  the  alcohol. 


52  EXPANSION   CURVE  OF   WATER.  [28 

28.     EXPANSION  CURVE  OF  WATER. 

The  variation  of  the  volume  of  a  given  mass  of  water,  as 
the  temperature  is  raised  by  steps  from  the  freezing  point,  is 
to  be  studied,  taking-  the  expansion  of  mercury  as  the  tem- 
perature standard.  It  should  be  remembered  that  our  choice 
of  a  thermometer  and  scale  of  temperatures  is  entirely  arbi- 
trary. The  statement  that  a  certain  substance  expands  "uni- 
formly" can  mean  only  that  it  expands  uniformly  with  the 
change  of  some  property  of  a  particular  substance  chosen  as 
a  standard.  Taking  the  expansion  of  mercury  as  a  standard, 
we  wish  here  to  determine  how  water  changes  in  volume  with 
change  of  temperature. 

(a)  The  bulb  of  the  water  thermometer  can  be  filled  by  the 
aid  of  a  reservoir.  The  reservoir  is  filled  with  warm  water 
and  the  end  of  the  thermometer  tube  introduced,  the  bulb 
being  below  the  reservoir.  The  bulb  is  then  alternately  heated 
to  drive  out  the  air  and  allowed  to  cool  to  admit  water.  Ask 
an  assistant  for  directions  if  there  is  difficulty.  Fill  the  ther- 
mometer until  the  water  stands  in  the  stem,  at  o°C.,  about 
2  cm.  above  the  bulb.  Fasten  the  tube  to  the  face  of  a  metric 
scale  and  place  the  bulb  in  the  water-bath.  The  bulb  is  first 
to  be  surrounded  with  shaved  ice.  When  conditions  become 
constant,  take  a  reading  of  the  height  of  the  water  meniscus 
and  also  of  the  mercury  thermometer  placed  in  the  bath  near 
the  bulb.  Melt  the  ice  and  gradually  raise  the  temperature 
of  the  bath  very  carefully,  at  first  reading  the  mercury  and 
water  thermometers  at  every  degree  between  o°  and  8°C., 
then  at  approximately  TO°,  15°,  20°,  and  every  ten  degrees 
thereafter  as  far  as  the  water  thermometer  will  permit. 

(b)  Determine  the  volume  of  the  water  in  the  bulb  at  o° 
C.  by  weighing  the  bulb  with  the  water  in  it  and  then  weigh- 
ing it  empty  and  dry. 

(c)  Determine  the   diameter  of  the  bore   either  by   direct 
measurement  with  the  micrometer  microscope  or,  better,  by 


28]  EXPANSION  CURVE  OF  WATER.  53 

placing  in  the  tube  a  thread  of  mercury,  measuring-  its  length 
and  then  weighing  the  mercury. 

((/)  From  the  determinations  of  the  volume  of  the  bull) 
and  the  diameter  of  the  bore  of  the  tube,  calculate  the  volume, 
in  cu.  cm.,  of  the  water  at  each  of  the  temperatures  observed, 
making  no  allowance  for  the  expansion  of  the  glass. 

On  coordinate  paper  plot  the  results  and  draw  a  curve,  hav- 
ing for  abscissae  the  temperatures  as  recorded  by  the  mercury 
thermometer,  and  for  ordinates  the  corresponding  volumes  of 
water.  In  doing  this,  choose  as  large  a  scale  for  volumes  as 
possible,  so  that  the  total  change  of  volume  will  about  cover 
the.  width  of  the  paper.  Since  your  observed  changes  of  vol- 
ume are  only  apparent  changes,  the  volume-change  of  glass 
must  be  added  in  order  to  obtain  the  true  expansion  of  the 
water.  To  do  this,  calculate  what  the  volume-increase  of 
the  water  thermometer  is  between  o°  and  ioo°C.,  due  to  the 
expansion  of  the  glass.  At  the  point  on  the  temperature  axis 
corresponding  to  100°  erect  an  ordinate  equal  to  this  expan- 
sion. Draw  a  slanting  line  through  your  origin  of  coordi- 
nates and  the  upper  end  of  this  ordinate.  The  lengths  of  the 
ordinates  between  this  line  and  the  temperature  axis  represent 
the  expansion  of  the  glass  for  the  corresponding  tempera- 
tures. Then,  from  various  points  along  the  apparent  expan- 
sion curve  of  water,  measure,  vertically  upward,  distances 
equal  to  the  volume-increase  of  the  glass  corresponding  to  this 
temperature.  Draw  a  smooth  curve  through  all  of  these 
points.  This  curve  referred  to  the  horizontal  axis  will  give 
the  true  expansion  of  the  water. 

(e)  State  any  conclusions  that  can  be  drawn  from  an  ex- 
amination of  the  curve  in  regard  to  the  behavior  of  water  as 
its  temperature  is  raised  from  o°  to  the  highest  point  reached. 

By  the  use  of  the  curve  determine  the  average  cubical 
coefficient  of  expansion  of  water  (i)  between  o°  and  ioo°; 
(2)  between  o°  and  20°,  (3)  between  o°  and  8°.  Also  deter- 
mine the  cubical  coefficient  of  expansion  of  water  at  15°. 


54         SPECIFIC  HEAT  OF  A  LIQUID  BY  METHOD  OF  HEATING.       29 

29.     SPECIFIC  HEAT  OF  A  LIQUID  BY  METHOD  OF 

HEATING. 

Reference.— Miller,  p.  188. 

In  this  experiment  a  heating  coil,  composed  of  high  resist- 
ance metal  through  which  an  electric  current  is  passed,  is 
immersed  for  a  given  time,  first  in  one  liquid  and  then  in 
another.  If  the  same  current  passes  through  the  coil  in  the 
two  cases,  equal  quantities  of  heat  should  be  generated  in 
equal  times.  Noting  the  mass  of  the  liquid  in  each  case  and 
the  rise  in  temperature,  the  two  quantities  of  heat  may  be 
equated  and  the  specific  heat  of  one  liquid  calculated,  if  that 
of  the  other  is  known.  Water,  taken  as  a  standard,  will  be 
one  liquid  used.  A  second  liquid,  properly  labeled,  will  be 
found  upon  the  laboratory  table.  The  method  is  applicable 
to  any  liquid  which  is  not  a  conductor  of  electricity  and  which 
does  not  act  chemically  upon  the  material  of  the  coil  or  calor- 
imeter. 

(a)  Place  the  bottle,  containing  the  liquid  to  be  used  in 
(b),  in  a  vessel  of  ice-water  to  cool.  Weigh  a  quantity  of 
ice-water  in  the  calorimeter  cup.  Set  up  the  calorimeter  and 
immerse  the  heating  coil,  having  the  temperature  of  the  water 
about  12°  below  the  room  temperature.  Allow  a  few  moments 
for  the  contents  of  the  cup  to  come  to  a  uniform  temperature, 
then  note  the  temperature,  and  turn  on  the  current  in  the  coil. 
Record  the  time  when  the  current  is  started,  and  also  the  time 
for  each  degree  rise  in  temperature  of  the  water  until  it 
reaches  a  temperature  as  far  above  that  of  the  room  as  it 
started  below.  Keep  the  water  well  stirred,  and  do  not  place 
the  thermometer  very  close  to  the  heating  coil. 

(&)  Repeat  (a),  using  the  liquid  furnished  instead  of 
water  in  the  calorimeter  cup.  Unless  known  from  previous 
work  it  will  be  necessary  to  find  the  water-equivalent  of  the 
calorimeter  cup,  stirrer,  and  thermometer. 


30]     SPECIFIC  HEAT  OF  A  LIQUID  BY  METHOD  0$  COOLING.     55 

(c)  Plot  on  the  same  sheet  of  paper  the  results  of  (a)  and 
of  (b),  using  temperatures  as  ordinates  and  times  as  abscissae. 
What  would  be  the  form  of  the  curve  for  each  case,  if  the  con- 
dition were  fulfilled  that  equal  quantities  of  heat  are  gained 
by  the  liquids  in  equal  times?    Compare  the  part  of  the  curve 
near  the  room  temperature  with  the  end  portions.     Erect  two 
perpendiculars  to  the  time  axis,  making  these  lines     cut     the 
curves  as  far  apart  as  possible.     From     these     intersections 
obtain  the  range  of  temperatures  passed  through  by  the  water 
and  the  other  liquid  in  equal  times.     Take     the     quantities 
gained  by  the  two  liquids  in  this  tim'e  as  equal,  and  form  an 
equation   from  which  the  specific  heat  of  the  liquid  may  be 
calculated.     From  the  result  just  obtained  calculate  what  mass 
of  the  liquid  will  be  "equivalent"  to  the  water  used  in   (a). 

(d)  If  you  have  time,  take  the  amount  of  liquid  found  in 
(c)  to  be  equivalent  to  the  water  used  in  (a),  and  repeat  (b). 
From  the  data  obtained  calculate  the  value  of  the  specific  heat. 
Why  should  this  value  be  more  reliable  than  the  one  found  in 

(0? 

30.     SPECIFIC  HEAT  OF  A  LIQUID  BY  METHOD  OF 

COOLING. 

References. — Millikan,   p.    206;    Miller,   p.    186;    Watson's    Practical 
Physics,  p.  224. 

This  method  is  a  comparison  of  the  quantities  of  heat  lost 
by  two  liquids,  one  of  which  is  water,  when  equal  volumes 
are  allowed  to  cool  under  exactly  the  same  conditions  through 
a  certain  range  of  temperature.  The  conditions  of  radiation 
being  the  same  for  both,  if  a  liquid  of  mass  ml  and  specific 
heat  Sj  cools  through  a  certain  temperature-range  in  Tx  sec- 
onds, and  a  second  liquid  of  mass  m2  and  specific  t  heat  sc 
requires  T,  seconds  for  the  same  temperature-change,  then 
the  quantities  of  heat  lost  will  be  proportional  to  the  times, 
i.  e.,  Q,/Q2  =  Tj/Ta.  If  w  denote  the  water-equivalent  of 


56        SPECIFIC  1I1CAT  01?  A  LIQUID  BY  METHOD  OF  COOLING.       [30 

the   containing   vessel,    thermometer,    and    stirrer,    the    above 
relation  becomes 

m^  -f-  w  _  T, 

m.,s2  +  w  ~"  T./ 
from  which  the  unknown  specific  heat  can  be  determined. 

(a)  A  large  jar  is  used,  having  a  wooden  cover  from  which 
is  suspended  a  smaller  vessel.     The  space  between  these   is 
made  a  water-jacket  by  putting  enough  water  in  the  larger 
so  that  when  the  cover  is  put  in  place  the  space  between  the 
two  vessels  will  be  filled  with  water.     The  liquid  used,  tur- 
pentine, is  now  heated  in  a  water-bath  to  about  85°  and  poured 
into  a  small  copper  cup,  closed  by  a  cork  through  which  the 
thermometer  and  stirrer  pass.  The  cup  is  then  passed  through 
the  wooden  cover  and  hangs  suspended  from  the  cork. 

Make  the  necessary  weighings  on  the  trip-scales.  Allow 
the  turpentine  to  cool  to  about  50°,  recording  the  time  for 
every  two  degrees  fall  in  temperature  at  first,  and  later  for 
each  one  degree  fall. 

(b)  Put   fresh   water  in  the  jacket,   and   repeat    (a)    with 
water  in  the   cup   instead  of  turpentine,   using  as  nearly   as 
possible  the  same  volume. 

(c)  Plot  the  cooling  curves  of  turpentine  and  of  water  on 
coordinate  paper,   using  temperatures   as   ordinates   and   cor- 
responding times  as  abscissae.     On  these  curves  take  a  cer- 
tain range  of  temperature  by  drawing  two  lines  parallel  to 
the  axis  of  abscissae,   each   line   cutting  both   curves.     Make 
this  temperature-interval  as  long  as  possible.  The  intersections 
of  these  lines  with  the  curves  will  give  the  times  required.  Cal- 
culate the  specific  heat  of  turpentine. 

If  the  water  had  not  been  changed  between  the  two  sets 
of  observations,  in  what  way  would  the  value  for  the  specific 
heat  of  tturpentine  have  been  affected  ?  What  source  of  error 
still  remains  even  if  the  water  in  the  jacket  is  changed  before 
the  second  measurement?  Suggest  a  way  to  avoid  this 
uncertainty. 


31  ]  MECHANICAL    EQUIVALENT   OE    HEAT.  57 

31.-     MECHANICAL    EQUIVALENT     OF     HEAT     BY 
METHOD  OF  PERCUSSION. 

References. — Preston,    p.    39;    Edser,    p.    269;    Watson,    p.    310. 

In  this  experiment  the  mechanical  equivalent  of  heat  is  to 
be  determined  from  the  work  done  by  a  falling-  body,  the  fall- 
ing taking  place  inside  a  bamboo  tube. 

(a)  Take  two  bottles,  each  containing  a  kilogram  of  shot, 
and  cool  the  shot  about  three  degrees  below  room  temperature 
by  placing  the  bottles  in  a  vessel  of  ice-water.  Pour  the  shot 
from  one  bottle  into  the  tube,  and  measure  inside  the  tube  to 
determine  the  mean  distance  the  shot  will  fall  when  the  tube 
is  turned  end  for  end.  To  correct  partially  for  radiation,  let 
the  first  trial  be  a  preliminary  one  to  determine  approximately 
what  the  rise  of  temperature  will  be.  Cool  the  shot  down 
below  the  room  temperature  far  enough,  so  that  when  it 
becomes  heated  it  will  rise  the  same  number  of  degrees  above 
room  temperature. 

To  determine  the  rise  in  temperature,  close  the  tube 
securely,  take  the  temperature  of  the  shot  by  inserting  the 
thermometer  through  the  small  hole  in  the  side  near  one  end, 
lay  the  tube  on  the  table,  and,  resting  one  end  on  the  table, 
raise  the  other  end  quickly  so  that  the  shot  does  not  fall  until 
the  tube  is  vertical.  Avoid  any  up  or  down  motion  of  the 
tube  as  a  whole.  Do  not  hold  the  tube  by  the  ends.  Let  the 
shot  fall  through  the  tube  100  times  in  this  way,  then  take 
the  temperature  again  through  the  small  hole.  Pour  the  shot 
back  into  the  bottle  and  cool  it  once  more.  Take  the  shot 
from  the  second  bottle,  and,  if  it  has  become  too  cool,  shake 
it  in  the  bottle.  Repeat  with  this  shot  while  the  tube 
is  still  warm. 

Make  four  or  five  measurements  in  this  way.  After  the 
tube  is  warm  it  may  be  necessary  to  take  some  other  initial 
temperature  to  properly  correct  for  radiation  as  suggested 
above. 


58  MECHANICAL    KOU1VAI.KNT    OF    H  KAT.  [32 

(b)  From  the  work  in  ergs  and  the  calories  of  heat  gained 
by  the  shot,  calculate  the  mechanical  equivalent  of  heat  for 
each  measurement.  Give  reasons  why  you  consider  some 
values  better  than  others. 

32.  MECHANICAL  EQUIVALENT  OF  HEAT  BY 
METHOD  OF  PULUJ. 

Reference. — Miller,  p.  194. 

This  method  of  determining  the  mechanical  equivalent  of 
heat  involves  the  measurement  of  the  work  done  in  raising  a 
given  mass  of  mercury  through  a  measured  range  of  tempera- 
ture. The  apparatus  consists  of  two  hollow  cones,  one  within 
the  other,  mounted  on  a  rotating  apparatus.  Mercury  is 
placed  in  the  inner  one  and  a  thermometer  suspended  in  the 
mercury.  The  outer  cone  is  rotated  and  tends  to  carry  the 
inner  cone  with  it.  This  is  prevented  by  a  lever  attached  to 
the  inner  cone.  From  one  end  of  the  lever  a  cord  passes 
over  a  pulley  to  a  scale-pan  on  which  a  mass  may  be  placed. 
The  deflection  of  the  lever  may  be  read  by  means  of  a  scale 
under  the  shorter  end.  If  the  outer  cone  is  rotated  with  con 
stant  angular  velocity,  the  pointer  of  the  lever  will  be  held  at 
a  constant  deflection.  The  force-moment  opposing  the  rotation 
of  the  inner  cone  may  then  be  measured.  Let  M  be  the  mass 
of  the  pan  and  its  contents,  and  f  the  friction  of  the  pulley. 
Then  the  total  force  acting  on  the  end  of  the  lever  is 
F  =  Mg  +  f  •  Let  L  be  the  length  of  the  force-arm  ( not  of 
the  wooden  lever).  The  force-moment  is  then  FL=  (Mg-f- 
f)L.  If  the  c.  g.  s.  units  are  used,  the  work  in  ergs  in  each 
revolution  is  W,  =  27rL(Mg  -f  f),  and  in  n  revolutions  is 

Wn  =  27rLn(Mg+f). 

In  making  n  revolutions,  let  us  suppose  that  the  temperature 
has  risen  from  t,  to  t2.  Let  the  sum  of  the  masses  of  the  two 
cones  be  mlt  and  the  specific  heat  of  the  steel  Sj.  Let  m2  be 
the  mass  of  the  mercury  in  the  inner  cone,  s,  its  specific  heat. 


OF 


321  MlvCHANICAI,    EQUIVALENT    OF    HEAT.  59 

and  w  the  water  equivalent  of  the  thermometer.  Then  the 
quantity  of  heat  in  calories  generated  is 

Q==  (nvs,  +  m2s2  +  w)[(t2  —  t,)  +R1, 

where  R  is  a  temperature  correction  to  allow  for  the  heat  lost 
by  the  cones  through  radiation  and  conduction  to  surrounding 
objects.  From  these  values  the  mechanical  equivalent  of  heat 
may  be  obtained  from  the  relation,  J  =  W/Q. 

(a)  The   masses   of  the    steel   cones   are   given.      Partially 
fill  the  inner  cone  with  mercury,  but  not  fuller  than  one  centi- 
meter below  the  top,  and  weigh.      Put  the  inner  cone,  ther- 
mometer, lever,  and  scale-pan     in  position.     Rotate  the  outer 
cone  and  adjust  the  mass    on    the    scale-pan  so    as    to    give 
a  steady  deflection  of  20  degrees  or  more.     When  the  lever  is 
deflected,  the  pulley  should  be  moved  so  that  the  cord  is  par- 
allel to  the  axis  of  the  machine. 

(b)  Read    the   temperature,    T1?    of   the   mercury,    remove 
the  thermometer  and  lever  (but  not  the  inner  cone),  and  per- 
form 200  rotations   with  the  same  speed  as  before.     Record 
the  final  temperature,  T2.     This  is  done  to  determine  the  rate 
at  which  the  temperature  is  changing,  due  to  radiation   and 
conduction,  before  the  test  is  made. 

(c)  Replace  the  lever  and  thermometer,  read  the  tempera- 
ture. t,,  and  rotate  the  cone  steadily  as  before  until  the  tem- 
perature has  risen  about  5°,  and  again  record  the  tempera- 
ture, t2.     Record  the  number  of  rotations,  n. 

(d)  Remove    the    lever    and    thermometer   and    again   per- 
form 200  rotations,  recording  the  temperatures,  T3  and  T4,  as 
in   (b).     This  enables  one  to  determine  the  rate  at  which  the 
temperature    is    changing,    due    to    radiation    and    conduction, 
after  the  test  is  made. 

(e)  Detach  the  cord  from  the  end  of  the  lever  and  attach 
masses   just   sufficient  to  balance  the   mass   of  the   scale-pan 
when  the  cord  is  hung  over  the  pulley.     Now  put  additional 


6O  COOIJXC,    THKOIV.H    C'HAXGK   Or    STATIC. 

masses  on  one  side  till  the  system  just  begins  to  move.  Call 
this  extra  mass  p ;  then  f  =  pg. 

(/)  From  (b]  the  fall  in  temperature  before  the  test,  due 
to  losses  during  one  rotation,  is  R,  =  ( T, — T.,)/2OO.  and 
from  (d)  the  loss  per  rotation  after  the  test  is  R2  =  (T3  — 
T4)/200.  Hence,  for  the  n  rotations  of  (r),  the  loss  due  to 
radiation  and  conduction  is 

R  =  -  —  (T,  —  T,  +  T,  —  T4). 

2   200   V 

Explain  fully  how  R  is  given  by  the  measurements  of  (b) 
and  (d)  and  the  last  calculation.  Calculate  J  from  the  equa- 
tion previously  given. 


33.     COOLING  THROUGH   CHANGE  OF  STATE. 

I.     Melting  Point  and  Cooling  Curve  of  Paraffine. 

(a)  Take  several  pieces  of  capillary  tubing,  each   2  or  3 
cm.  long,  dip  the  ends  in  melted  paraffine,  and  let  them  fill  by 
capillarity.     Fasten  them  around  the  bulb  of  a  thermometer 
by  means  of  a  rubber  band.     Immerse  the  thermometer  bulb 
in  a  test-tube  of  water  and  heat  this  in  a  water-bath.    Note  the 
temperature  at  which  the  paraffine  melts.     Remove  the  test 
tube  containing  the  thermometer  and  note  the  temperature  at 
which  the  paraffine  solidifies.     Take  the  mean   of  these  two 
as  the  melting  point. 

(b)  Put  a  thermometer  in  a  test-tube  together  with  enough 
paraffine  to  cover  the  bulb  when  melted.     Heat  the  test-tube 
in   a   water-bath   until   the  thermometer   registers   about   70°. 
Remove  the  test-tube  from  the  bath,  clamp  it  in  a  stand,  and 
allow  the  paraffine  to  cool  slowly  in  air  to  about  38° C.    Record 
the  time   for  each   degree  or  half -degree  fall,   or   at   shorter 
intervals  when  the  cooling  is  evidently  not  uniform.     Plot  on 
coordinate  paper,  using  temperatures  as  ordinates  and  times 


34 1  MAT  OF  FUSION.  61 

as  abscissae.     Explain  the  form  of  the  different  parts  of  the 
curve. 

II.     Cooling  Curve  of  Acetamide. 

Place  a  thermometer  in  a  test-tube  and  surround  the  ther- 
mometer with  crystals  of  acetamide,  filling  the  test-tube  about 
one-third  full.  Place  the  test-tube  in  a  water-bath  and  heat 
to  the  temperature  of  boiling  water.  Remove  the  test-tube 
from  the  bath,  clamp  it  in  a  stand,  and  either  note  the  time 
for  each  degree  or  half-degree  of  fall,  or  record  the  tempera- 
ture every  half-minute.  Continue  the  readings  until  the  tem- 
perature falls  to  about  40°.  Plot  on  coordinate  paper,  using 
temperatures  as  ordinates  and  times  as  abscissae.  Explain  the 
form  of  the  different  parts  of  the  curve. 

34.     HEAT  OF  FUSION. 
References. — Watson,  p.  246;  Ferry  and  Jones,  p.  231. 

The  substance  whose  heat  of  fusion  it  is  desired  to  measure 
is  an  alloy  known  as  Wood's  fusible  metal.  Its  composition 
is  lead  15.9  parts,  cadmium  7  parts,  bismuth  52.4  parts,  and 
tin  14.2  parts.  The  alloy  is  a  solid  at  ordinary  temperatures, 
but  readily  melts  in  hot  water.  The  method  employed  will 
be  the  method  of  mixtures.  A  known  mass  of  the  metal  is 
placed  in  a  nickel  crucible  of  known  mass  and  specific  heat, 
and  is  heated  to  ioo°C.  The  whole  is  then  plunged  into  a 
known  quantity  of  cold  water  in  a  calorimeter  cup  and  the 
change  in  temperatures  noted.  From  these  data,  if  the  spe- 
cific heat  of  the  metal  in  the  solid  and  in  the  liquid  state  be 
known,  the  heat  of  fusion  may  be  found.  Let  M  be  the  mass 
of  the  alloy;  m  the  mass  of  the  nickel  crucible;  W  the  mass 
of  the  water  in  the  calorimeter  cup  plus  the  water-equivalent 
of  the  cup,  thermometer,  and  stirrer;  s,_  the  specific  heat  of 
the  liquid  alloy ;  s2  that  of  the  solid  alloy ;  s3  that  of  the  nickel 
crucible;  T  the  melting  point  of  the  alloy;  t±  the  initial  tern- 


62  HEAT  OF  FUSION.  [34 

perature  of  the  alloy  and  crucible ;  t0  the  initial  temperature  of 
the  calorimeter,  t  the  final  temperature;  and  L  the  heat  of 
fusion  of  the  alloy.  Write  the  proper  equation  representing 
the  transfers  of  heat  in  the  above  process,  using-  the  symbols 
indicated,  and  solve  the  equation  for  L. 

(a)  First  determine  the  mass,   in  grams,  of  those   things 
whose  mass  it  is  necessary  to  know.     Place  the  alloy  in  the 
crucible  and  determine,  within  3°,  the  melting  point  of  the 
alloy.     To  do  this,  stand  the  crucible   (in  a  clamp  provided 
for  it)  in  a  vessel  of  water.     Heat  the  water,  taking  care  after 
the  water  has  reached  the  temperature  of  60°  that  the  heating 
be  done  slowly,  so  that  the  metal  will  be  at  the  same  tempera- 
ture as  the  water.     The  thermometer  must  be  placed  in  the 
water  and  not  in  the  metal.     The  liquid  metal  "wets"  glass, 
hence  metal  would  be  withdrawn  with  the  thermometer  and 
relatively  large  changes  in  mass  introduced.     If  the  thermom- 
eter were  placed  in  the  metal  and  left  there,  there  would  be 
great  danger  of  its  breaking  on  the  solidification  of  the  metal. 
After  the  melting  point  has  been  found,  bring  the  water  to 
the  boiling  point,  taking  care  that  no  water  gets  inside  the 
crucible.     Remove  the  crucible,  quickly  wipe  the  outside,  and 
carefully  drop  it  in  the  calorimeter,   right  side  up,  with  its 
contained  metal,  letting  the  water,  if  it  will,  run  into  the  cru- 
cible and  thus  more  quickly  cool  the  metal.     Note  the  time 
taken  for  the  temperature  to  become  uniform,  then  note  the 
rate  of  cooling  and   correct   for   radiation.     When   cold,   the 
metal  can  be  dumped  out  of  the  crucible,  leaving  the  crucible 
clean. 

The  specific  heats  of  the  solid  and  liquid  alloy  will  be  given, 
or  if  time  permits  they  may  be  found  by  the  method  of  mix- 
tures. The  specific  heat  of  nickel  may  be  found  in  a  book  of 
Tables. 

Make  two  or  three  determinations,  as  outlined  above,  of 
the  heat  of  fusion  of  Wood's  alloy. 

(b)  Why  is  only  a  rough     determination     of  the  melting 


35 1  HEAT  OF  VAPORIZATION   AT   BOILING   POINT.  63 

point  necessary?  Discuss  the  relative  accuracy  with  which 
the  different  masses  used  must  be  determined  for  a  given 
accuracy.  Point  out  the  sources  of  error  in  the  experiment. 

35.     HEAT  OF  VAPORIZATION  AT  BOILING  POINT. 

References. — Edser,  p.   150;  Watson's  Practical  Physics,  p.  237. 

In  this  experiment  Kahlenberg's  modification  of  Berthe- 
lot's  apparatus1  is  used. 

(a)  Determine  the  boiling  point  of  the  liquid  used,  by  care- 
fully  heating   a   small   quantity   in   a   test-tube   or   beaker  by 
means  of  a  water-bath. 

(b)  Weigh  the  calorimeter,  first  dry  and  empty,  then  about 
two-thirds  full  of  water.     Carefully  dry  and  weigh  the  worm, 
together  with  the  two  corks  which  fit  its  ends.     Set  up  the 
calorimeter  with  stirrer,  worm,  and  thermometer.     The  boiler 
consists  of  a  test-tube  to  which  is  fitted  a  rubber  stopper.     A 
glass  tube  extends  through  the  stopper  to  the  bottom  of  the 
test-tube;  two  wires  also  pass  through  the  stopper,  and  are 
connected  to  a  coil  of  wire  which  loosely  surrounds  a  part 
of  the   glass   tube.      When   in   use   the  test-tube  is   inverted, 
enough  liquid  being  placed  in  it  to  completely  cover  the  coil 
of  wire  after  the  tube  is  inverted.     An  electric  current  is  then 
sent  through  the  coil,  furnishing  the  heat  to  boil  the  liquid. 
The  vapor  from  the  boiling  liquid  passes  downward  through 
the  glass  tube  and  enters  the  worm,  when  the  boiler  is  placed 
in  position  over  the  calorimeter. 

Care  should  be  taken  to  use  enough  liquid  so  that  the  heat- 
ing coil  is  covered  throughout  the  experiment.  Never  allow 
the  heating  current  to  be  closed  through  the  coil  while  the 
coil  is  not  completely  covered  with  liquid.  Do  not  place  the 
boiler  over  the  calorimeter  until  the  liquid  boils  and  the  vapor 
is  issuing  freely  from  the  tube.  See  that  the  cork  is  removed 

journal  of  Physical  Chemistry,   1901,  Vol.  5,  p.  215.  . -, 


64  HEAT  OF  VAPORIZATION   AT  ROOM   TEMPERATURE.  [36 

from  the  free  end  of  the  worm,  as  the  boiling  must  be  done 
at  atmospheric  pressure,  otherwise  the  temperature  of  the 
vapor  will  not  be  that  found  in  (a). 

When  all  is  ready,  note  the  temperature  of  the  calorimeter, 
and  place  the  boiler  in  its  proper  place  so  that  the  vapor  enters 
the  worm.  Gently  stir  the  water  in  the  calorimeter,  and  read 
the  thermometer  at  one-minute  intervals  until  the  temperature 
has  risen  about  5°.  Turn  off  the  current,  remove  the  boiler, 
cork  the  ends  of  the  worm,  and  continue  to  read  the  ther- 
mometer at  one-minute  intervals  for  five  minutes.  Remove 
the  worm  from  the  calorimeter,  carefully  dry  the  outside,  and 
weigh.  Pour  the  contents  of  worm  and  boiler  into  the  proper 
bottle,  and  empty  the  calorimeter.  See  that  the  electric  cir- 
cuit is  disconnected. 

(c)  From  the  series  of  temperatures  taken  determine  the 
rise  of  temperature  of  the  calorimeter,  correction  being  made 
for  radiation.  Determine  the  water-equivalent  of  the  calor- 
imeter and  contents,  including  the  stirrer,  thermometer  (if  the 
equivalent  of  the  thermometer  is  not  known,  it  can  be  found 
by  the  method  given  in  Exp.  36),  empty  worm,  and  water. 
The  necessary  specific  heats  may  be  obtained  from  the  Tables. 
Calculate  the  amount  of  heat  gained  by  the  calorimeter. 
Knowing  the  mass  of  the  vapor  condensed,  the  change  in 
temperature  of  the  liquid,  and  the  specific  heat  of  the  liquid 
(see  Tables  for  the  specific  heat),  calculate  the  heat  transferred 
to  the  calorimeter,  and  determine  the  heat  of  vaporization  of 
the  liquid  at  its  boiling  point. 

36.     HEAT  OF  VAPORIZATION  AT  ROOM 
TEMPERATURE. 

The  heat  of  vaporization  of  a  liquid  varies  with  the  temper- 
ature at  which  vaporization  takes  place.  In  nature  vaporiza- 
tion takes  place,  for  the  most  part,  at  atmospheric  tempera- 
ture rather  than  at  boiling  temperature.  The  object  of  this 


36]  HEAT  OF  VAPORIZATION   AT  ROOM   TEMPERATURE.  65 

experiment  is  to  find  the  amount  of  heat  necessary  to  vapor- 
ize one  gram  of  a  liquid  at  the  room  temperature.  To  do 
this,  dry  air  is  made  to  bubble  through  the  liquid,  thus 
increasing  the  free  surface  and  producing  rapid  evaporation. 
The  loss  of  weight  of  the  liquid  gives  the  amount  evaporated, 
while  from  the  fall  of  temperature  of  the  liquid  and  calorim- 
eter, together  with  their  masses  and  specific  heats,  the  heat- 
loss  can  be  determined  and  the  heat  of  vaporization  calcu- 
lated. 

(a)  Carefully    weigh    the   calorimeter   cup    when    dry   and 
empty,  and  again  when   containing  about    100  grams   of  the 
liquid  whose  heat  of  vaporization  is  desired.     Place  the  cover 
on  the  calorimeter,  writh  the  thermometer  bulb  in  the  liquid 
and  arranged  so  that  dry  air  can  be  forced  through  the  liquid 
by  means  of  a  small  foot-bellows.     Have  the  initial  tempera- 
ture of  the  liquid  2°  or  3°  above  the  room  temperature.     Pass 
the  dry  air  through  the  liquid,  allowing  ample  room  for  the 
vapor-charged  :air  to  escape,  until  the  temperature  is  as  much 
below  room  temperature  as  the  initial  temperature  was  above 
room   temperature.      Weigh   the   calorimeter     and    remaining 
liquid.     A  50°  thermometer  graduated  in  o°.i  should  be  used. 
Wet  the  thermometer,  with  the  liquid  used,  about  as  high  as 
the  depth  to  which  it  will  be  placed  in  the  liquid  in  the  calor- 
imeter, so  that  as  much  liquid  will  be  introduced  at  first  as 
will  be  withdrawn   later  when  the  thermometer  is   removed 
from  the  calorimeter. 

(b)  Repeat  the  work  of  (a),  this     time     drawing    dry  air 
through  the  calorimeter  by  means  of  a  jet-pump.  When  finished, 
empty  and  dry  the  calorimeter.     If  a  liquid  other  than  water 
was  used,  the  liquid  should  be  poured  back  into  its  proper 
bottle. 

(c)  From  the  amount  of  liquid  evaporated,  the     fall     in 
temperature,    and   the    water-equivalent    of   the   thermometer, 
calorimeter,  and  liquid  used,  determine  the  heat  of  vaporiza- 
tion in  (a)  and  (b),  taking  the  mean  as  the  final  value.     The 


66  FREEZING  POINT  OF  SOLUTIONS.  [37 

amount  of  liquid  cooled  may  be  taken  as  the  mean  of  the 
initial  and  final  amounts. 

(d)   Point  out  the  chief  sources  of  error. 

What  reason  can  you  suggest  for  the  increase  of  the  heat 
of  vaporization  of  a  liquid  as  the  temperature  of  the  liquid  is 
decreased  ? 

Evaporation  takes  place  from  dry  ice  at  temperatures 
below  the  freezing  point.  This  change  from  solid  to  vapor 
is  called  sublimation.  By  what  amount  would  you  expect  the 
heat  of  sublimation  to  exceed  the  heat  of  vaporization  for 
any  given  substance? 

37.     FREEZING  POINT  OF  SOLUTIONS. 

References. — Watson,   p.  268;  Watson's   Practical   Physics,   p.   258; 

Edser,   p.    167. 

The  object  of  this  experiment  is  to  observe  the  lowering  of 
the  freezing  point  of  water  caused  by  dissolving  salt  and 
sugar  in  it  to  form  solutions  of  different  concentrations,  and 
to  determine  the  molecular  weights  by  means  of  this  low- 
ering. 

(a)  Using   a   50°    thermometer,   determine     the      freezing 
point  of  pure  water  with  the  same  apparatus  as  that  employed 
in  the  calibration  of  the  100°  thermometer.  Then  determine  the 
freezing  point  of  a  4  per  cent  solution  of  common  salt  in  water. 
By  percentage  solution  is  here  meant  the  number  of  grams  of 
dissolved  substance  per   100  grams  of  the  solution.     Repeat 
for  an  8  per  cent  and  for  a  12  per  cent  solution. 

(b)  Repeat    (a)    with   aqueous   solutions   of   sugar  of    12, 
20,  and  40  per  cent  concentration. 

(c)  Tabulate  the  results  of  (a)  and  of  (b),  and  for  each 
case  calculate  the  lowering,  per  gram  of  dissolved  substance, 
of  the  freezing  point  of  a  given  mass  of  water.     What  rela- 
tion seems  to  hold  between  the  change  of  freezing  point  of  a 
given  mass  of  water  and  the  mass  of  dissolved  substance? 


38]  HKAT  OF  SOLUTION.  67 

Note  the  difference  of  freezing  point  for  12  per  cent  solu- 
tions of  salt  and  sugar. 

Calculate  the  molecular  weights  of  salt  and  of  sugar  from 
the  relation,  M  =  Ks/St,  where  s  is  the  number  of  grams 
of  dissolved  substance,  S  is  the  number  of  grams  of  the  sol- 
vent, t  is  the  depression  of  the  freezing  point,  and  K  is  1850 
for  aqueous  solutions. 


38.     HEAT  OF  SOLUTION. 

The  quantity  of  heat  absorbed  in  the  solution  of  one  gram 
of  a  substance  in  a  large  amount  of  the  solvent  is  called  its 
heat  of  solution.  If  heat  is  given  out  in  the  solution,  the 
quantity  is  considered  negative. 

If  the  temperature  of  the  salt  after  solution  be  different 
from  that  at  which  it  was  poured  into  the  water,  it  will  be 
necessary  to  consider  its  specific  heat  also.  According  to 
the  following  method  the  heat  of  solution  and  the  specific 
heat  are  both  determined,  although  the  former  is  the  main 
object  of  the  experiment. 

(a)  Calculate  the  thermal  capacity  of  the  calorimeter  cup 
and  stirrer  from  their  masses  and  specific  heats,  the  latter  of 
which  may  be  found  in  a  book  of  Tables.  To  this  must  be 
added  the  thermal  capacity  of  the  immersed  portion  of  the 
50°  thermometer  used.  This  may  be  experimentally  deter- 
mined with  sufficient  accuracy  as  follows :  Set  up  the  cal- 
orimeter with  water  at  the  room  temperature  in  both  cup  and 
jacket,  having  first  weighed  the  cup  and  contained  water. 
Record  the  temperature  when  it  has  become  steady,  then  take 
out  the  thermometer  and  immerse  it,  to  the  same  depth  as 
usual  in  the  cup,  in  water  at  about  45°.  After  a  few  min- 
utes read  the  temperature  to  o°.i,  then  as  quickly  as  possible 
take  out  the  thermometer  and  put  it  into  the  usual  position 
in  the  calorimeter,  shaking  off  superfluous  water  on  the  way. 


68  1 1  HAT  OF  SOLUTION.  [38 

Stir  until  the  temperature  becomes  steady  and  record  the 
reading.  Then  calculate  the  thermal  capacity  of  the  ther- 
mometer. 

(6)  On  one  of  the  Becker  balances  weigh  out  on  pieces 
of  dry  paper  two  portions  of  salt,  each  of  about  15  grams,  to 
o.oi  gram.  Make  sure  that  the  salt  is  quite  dry  and  finely 
pulverized,  and  be  careful  not  to  leave  any  in  the  balance-pan. 
This  amount  of  salt,  if  sodium  hyposulphite  be  used,  when 
dissolved  in  200  grams  of  water  will  lower  its  temperature 
a  little  over  3°.  It  is  best  to  have  the  cup  about  3°  warmer 
than  the  jacket,  because  the  larger  part  of  the  salt  dissolves 
in  a  few  seconds,  so  that  the  loss  of  heat  by  radiation  during 
this  time  is  small ;  and  the  temperature  being  then  reduced 
to  about  that  of  the  jacket,  there  is  no  loss  by  radiation  dur- 
ing the  longer  time  required  for  the  complete  solution  of  the 
salt. 

(c)  Set    up   the   calorimeter,    with    the    jacket    filled    with 
water  at  the  room  temperature,  and  the  cup  containing  200 
grams  of  water  about  3°  warmer.     Keep  the  stirrer  moving 
slowly  and  read  the  temperature  of  the  cup  at  intervals  of 
one  minute   for   about   five   minutes.      Pour   in  the   salt   one 
minute   after  the   last  observation,   stir   rather   vigorously  to 
hasten  solution,  and  record  the  final  temperature. 

From  the  series  of  observations,  calculate  the  temperature 
of  the  cup  at  the  time  when  the  salt  was  poured  in.  The  tem- 
perature of  the  salt  at  that  time  may  be  assumed  to  be  that 
of  the  room. 

(d)  Make  a   similar  trial  with   a   second   portion  of  salt, 
having  the  cup  at  about  40°  C.     Make  sure  that  there  is  the 
proper   difference   between   cup   and   jacket   at   the   time   the 
salt  is  poured  in. 

(e)  Call  the  specific  heat  of  the  salt  x,  and  the  heat  of 
solution  y.     Write  for  each  case  an  equation   involving  the 
following  quantities : 

i.     Heat  lost  by  water  in  cup. 


39]  HEAT  OF  CHEMICAL  COMBINATION.  69 

2.  Heat  lost  by  thermometer,  cup,  and  stirrer. 

3.  Heat  gained  by  salt  in  changing  temperature. 

4.  Heat  absorbed  during  solution  of  salt. 

Solve  the  two  equations  for  x  and  y.  Why  does  this  method 
give  widely  varying  results  for  the  specific  heat,  while  the 
results  for  heat  of  solution  are  fairly  consistent? 

Caution : — Do  not  leave  the  solution  standing  in  the  cup. 
Wash  it  out  as  soon  as  possible. 

39.     HEAT  OF  CHEMICAL  COMBINATION. 

Determination  of  the  heat  generated  by  the  combination  of 
sodium  hydroxide  with  hydrochloric  acid  to  form  sodium 
chloride. 

A  0.5  normal  solution  of  each  of  the  above  compounds  is 
furnished.  By  a  normal  solution  is  meant  one  which,  in  1000 
grams  of  the  solution,  contains  a  mass  of  the  element  (which 
is  to  enter  into  the  new  combination)  equal  in  grams  to  its 
atomic  weight.  Thus  the  normal  solution  of  sodium  hydrox- 
ide is  a  solution  which  contains,  in  looo  grams  of  the  solution, 
40  grams  (23-}-  T6  +  i)  of  sodium  hydroxide,  or  23  grams 
of  sodium.  The  0.5  normal  solution  contains  one- 
half  as  much  sodium  in  the  same  quantity  of  solution. 

It  is  evident  that  if  equal  masses  of  these  solutions  be  mixed, 
the  reaction  will  be  just  completed,  and  the  result  will  be 
a  neutral  solution  of  sodium  chloride.  The  solutions  are 
to  be  mixed  in  the  calorimeter  cup  at  as  nearly  as  possible 
the  same  temperature,  and  the  resulting  rise  of  temperature 
noted.  The  alkali  should  be  placed  in  the  cup,  and  the  acid 
added  to  it.  The  acid,  being  immediately  neutralized,  will 
then  have  no  action  on  the  metal  of  the  cup. 

(a)  Weigh  out  100  grams  of  the  sodium  hydroxide  solution 
in  the  cup,  and  the  same  amount  of  the  hydrochloric  acid  solu- 
tion in  the  beaker.  The  latter  should  be  weighed  out  roughly  at 
first,  poured  back  into  the  bottle,  then  the  wet  beaker  counter- 


70  EXPANSION  OF  A  GAS  BY  FLASK  METHOD.  [40 

poised  and  the  amount  weighed  accurately.  This  will  then 
be,  fairly  accurately,  the  mass  which  is  afterward  poured 
into  the  calorimeter.  A  small  error  is  introduced  by  taking 
the  second  thermometer  out  of  the  beaker  after  reading  its 
temperature,  but  this  may  be  neglected. 

If  care  has  been  taken  not  to  handle  the  cup  and  beaker 
any  more  than  is  necessary,  the  two  temperatures  should  be 
very  nearly  the  same  when  ready  for  use.  Since  the  amounts 
used  are  equal,  it  may  be  safely  assumed  that  the  resulting 
solution  of  sodium  chloride  has  risen  to  the  final  temperature 
from  the  mean  of  the  two  initial  temperatures. 

A  direct  determination  of  the  specific  heat  of  the  sodium 
chloride  solution  is  impracticable.  The  value,  0.987,  which 
has  been  calculated  by  interpolation  from  tabulated  results, 
may  be  used  for  this  case. 

Make  two  trials,  and  calculate  for  each  the  quantity  of 
heat  which  would  have  been  evolved  if  1000  grams  of  nor- 
mal solution  had  been  used  in  each  case. 

(b)  Repeat  the  work,  if  there  is  time,  with  similar  solu- 
tions of  potassium  hydroxide  and  sulphuric  acid,  and  com- 
pare the  results. 

40.     COEFFICIENT  OF  EXPANSION  OF  A  GAS  AT 
CONSTANT  PRESSURE  BY  FLASK  METHOD. 

(a)  Thoroughly  dry  the  flask  or  bulb  by  rinsing  out  ten 
times  or  more  with  dry  air,  then  hang  it  in  the  boiler  with 
the  bulb  down  and  with  the  stop-cock  open.  If  there  is  any 
chance  for  the  steam  to  enter,  attach  a  rubber  tube  to  the 
open  end  and  place  the  other  end  of  this  tube  where  the  steam 
cannot  enter.  Boil  the  water,  causing  the  steam  to  pass  around 
the  whole  flask  until  the  air  inside  is  at  the  temperature  of  the 
steam.  Then  close  the  stop-cock,  remove  from  the  boiler, 
and  allow  to  cool.  Next  place  it  under  the  surface  of  the 
ice-water  until  it  has  assumed  that  temperature.  Then  open 


41  ]  CONSTANT-PRESSURE  AIR-THERMOMETER.  71 

the  stop-cock  under  water,  allowing  the  water  to  enter  but 
not  the  air  to  escape.  Raise  the  bulb  so  that  the  level  of  the 
water  inside  is  the  same  as  that  without,  thus  assuring  the 
same  pressure.  Close  the  stop-cock,  remove  and  dry,  and  then 
carefully  weigh.  In  order  to  obtain  the  volume,  the  flask  must 
now  be  weighed  full  of  water,  and  then  again  empty  and  dry. 
It  is  best  to  fill  with  ice-water  and  to  make  the  weighings 
when  it  is  cold,  so  as  to  get  the  volume  at  o°C.  In  drying 
the  flask,  great  care  should  be  taken  not  to  break  the  stop- 
cock by  the  heat.  These  weighings  will  enable  you  to  deter- 
mine the  volume  of  the  air  in  the  flask  when  under  atmos- 
pheric pressure  and  at  o°C. 

From  the  results  of  the  above  measurements  and  the 
coefficient  of  expansion  of  glass  find  the  coefficient  of  expan- 
sion of  air. 

(b)  Repeat  with  some  available  gas  other  than  air,  and 
compare  the  result  with  that  of  air. 

41.     COEFFICIENT  OF  EXPANSION  OF  AIR.     CON- 
STANT-PRESSURE AIR-THERMOMETER. 

References. — Edser,  p.   108;  Watson's  Practical  Physics,  p.  209. 

The  object  of  the  experiment  is  to  study  the  variation  of 
the  volume  of  a  gas  when  heated  under  constant  pressure 
(Gay  Lussac's  Law),  and  to  determine  the  average  coeffi- 
cient of  cubical  expansion  of  air  between  o°  and  ioo°C. 
In  the  form  of  constant-pressure  air-thermometer  used,  the 
air  (carefully  dried)  is  contained  in  a  glass  tube  graduated 
in  cu.  cm.  and  closed  at  one  end.  The  graduated  tube  is  con- 
nected to  an  open  glass  tube  by  rubber  tubing,  forming  a 
"U"  containing  mercury.  The  pressure  on  the  enclosed  air 
can  be  regulated  by  raising  or  lowering  the  open  glass  tube. 
Surrounding  the  graduated  tube  containing  the  air  is  a  ves- 
sel, covered  by  an  asbestos  jacket,  in  which  a  bath  of  water 


72  CONSTANT-PRESSURE  AIR-THERMOMETER.  [4! 

may  be  placed  or  through  which  steam  may  be  passed.  The 
graduated  tube  is  vertically  adjustable  through  a  sleeve  in 
the  bottom  of  the  vessel,  so  that  the  meniscus  of  the  mercury 
may  be  seen  outside  and  the  volume  read. 

(a)  Fill  the  vessel  with  a  mixture  of  ice  and  water,  and, 
when  the  enclosed  air  has  had  time  to  come  to  the  tempera- 
ture of  the  bath,  read  the  volume,  after  adjusting  the  pres- 
sure so  that  it  is  equal  to  atmospheric  pressure. 

Fill  the  vessel  with  water  at  io°C.,  adjust  the  pressure,  and 
again  read  the  volume.  In  this  way  raise  the  temperature 
by  steps,  reading  the  volume  of  the  air  at  10°,  20°,  30°,  45°, 
60°,  80°,  taking  care  each  time  to  wait  long  enough  (three  min- 
utes or  more)  for  the  enclosed  air  to  come  to  the  same  tem- 
perature as  the  bath,  and  each  time  adjusting  the  pressure 
so  that  it  is  equal  to  atmospheric  pressure.  The  mercury 
meniscus  on  the  closed-tube  side  should  always  be  as  close 
to  the  bottom  of  the  jacket  as  will  permit  of  reading  the 
level. 

After  closing  the  top,  pass  steam  througn  the  vessel,  in 
at  the  bottom  and  out  at  the  top,  and  take  another  reading 
of  the  volume,  pressure  conditions  being  the  same  as  before. 
It  will  probably  be  necessary  to  wait  longer  in  this  case  than 
in  the  other  cases  for  the  air  to  reach  the  temperature  of 
the  steam. 

(b)  Make  another  and  similar  series  of  observations  at  a 
pressure  10  cm.  above  atmospheric  pressure. 

(r)  Plot  the  observations  of  (a)  and  of  (b)  on  the  same 
sheet  of  coordinate  paper,  using  temperatures  (centigrade) 
as  abscissae  and  volumes  as  ordinates. 

From  the  volume  at  o°C.  and  the  volume  at  100° 
C.,  as  taken  from  the  curve,  calculate,  for  each 
curve,  the  average  apparent  coefficient  of  expansion  of  the 
air  between  those  temperatures.  Take  the  mean  of  the  two 
results,  correct  for  the  expansion  of  the  glass,  and  obtain  /?. 
the  absolute  coefficient  of  expansion  of  air. 


42]  CONSTANT-VOLUME    AIR-THERMOM£TER.  73 

(d)  Show  how  the  volume  varies  with  the  absolute  tem- 
perature, the  pressure  being  constant. 

Taking  some  particular  volume  on  the  two  curves,  com- 
pare the  corresponding  temperatures  and  pressures.  Do  the 
same  for  another  volume.  Knowing  the  pressure  corre- 
sponding to  each  of  the  two  curves,  how  does  the  pressure 
vary  with  the  temperature  when  the  volume  is  constant? 

Taking  the  same  temperatures  on  the  two  curves,  find  how 
the  volume  varies  with  the  pressure. 

(e)  Write  an  equation   connecting  the  temperature,   pres- 
sure, and  volume  of  a  gas  which  might  be  inferred  from  the 
three  parts  of  (d). 

42     CONSTANT- VOLUME     AIR-THERMOMETER. 

References. — Watson's   Practical   Physics,   p.   203;   Millikan,   p.    125 
and  p.  131;  Miller,  p.   166. 

The  object  of  this  experiment  is  to  study  the  law  of  vari- 
ation of  the  pressure  of  a  given  quantity  of  enclosed  air  at 
constant  volume  as  the  temperature  is  changed,  and  also  to 
determine  the  pressure  coefficient  of  the  gas.  The  air  is 
enclosed  in  a  glass  bulb  placed  inside  a  vessel  so  that  the 
bulb  may  be  surrounded  by  a  water-bath,  by  shaved  ice,  or 
by  steam.  A  thermometer  is  placed  in  the  bath  to  determine 
its  temperature.  The  pressure  on  the  enclosed  gas  is  regu- 
lated by  means  of  a  mercury  column  in  a  rubber  tube,  con- 
necting on  the  one  side  with  the  glass  tube  which  forms 
an  extension  of  the  bulb,  and  on  the  other  with  an  open  ver- 
tical glass  tube  whose  position  is  vertically  adjustable.  The 
pressure  on  the  enclosed  air  may  be  determined  from  the  dif- 
ference in  level  of  the  mercury  on  the  two  sides  and  the 
barometric  reading.  The  volume  of  the  air  in  the  bulb  is  made 
the  same  before  each  reading  by  bringing  the  mercury  men- 
iscus in  contact  with  a  glass  point  inside  the  glass  tube 
attached  to  the  bulb. 


74  CONSTANT-VOLUME    AIK-THICRMOMETKR.  |  4-' 

Caution : — The  mercury  on  the  bulb  side  should  always 
be  lowered  some  distance  before  changing  to  a  lower  tem- 
perature. Be  especially  careful  to  do  this  before  removing 
the  steam  when  you  have  taken  a  reading  at  the  boiling 
point.  (Otherwise,  on  cooling,  the  mercury  will  run  into 
the  bulb.)  Do  not  hurry  in  taking  the  readings  after  chang- 
ing the  temperature,  but  wait  until  the  meniscus  set  at  the 
glass  point  does  not  move. 

(a)  Without  any  bath  in  the  reservoir,  while  all  is  at  the 
room  temperature,  bring  the  mercury  to  the  glass  point  and 
determine  the  pressure.     Record  the  room  temperature,  and 
the  barometric  reading. 

(b)  After  having  lowered  the  mercury  on  the  bulb  side, 
surround  the  bulb  with  shaved  ice,   and  then  determine  the 
pressure  with  the  meniscus  at  the  glass  point.     The  tempera- 
ture may  be  taken  as  o°C. 

Melt  the  ice,  and  then  make  a  series  of  determinations  of 
the  pressure  with  the  water  in  the  vessels  having  in  turn 
temperatures  of  10°,  20°,  30°,  45°,  60°,  and  80°  (approxi- 
mately). 

Pass  steam  through  the  vessel,  making  another  determin- 
ation. This  temperature  may  be  found  by  determining  the 
boiling  point  from  the  known  atmospheric  pressure. 

Arrange  all  observations  in  tabular  form. 

(c)  Plot   on   coordinate   paper   the   results   of    (b),    using 
temperatures    as    abscissae    and    the   corresponding    pressures 
as  ordinates.     Draw  a  smooth  curve  through  the  points  of 
the  plot. 

Calculate  the  mean  increase  of  pressure  per  degree  increase 
in  temperature  from  o°  to  100°,  using  values  taken  from 
the  plot.  This  is  the  temperature  coefficient  (ft)  of  pressure  of 
a  gas.  Write  it  as  a  decimal  and  find  its  reciprocal.  The 
negative  of  this  represents  what  point  on  the  absolute  scale 
of  temperatures  ? 


43]  VARIATION    OF   A    SATURATED    VAPOR. 


75 


(d)  Write  an  equation  connecting  P0,  the  pressure  at  o°  ; 
P,  the  pressure  at  t°  ;  t ;  and  j3. 

Using  this  equation  and  the  pressure  of  (a),  calculate  the 
temperature  of  the  room,  thus  using  the  apparatus  as"  a  ther- 
mometer. Compare  the  result  with  the  room  temperature 
as  read  from  a  mercury  thermometer. 

(e)  Show  from  your  results  how  the  pressure  of  the  gas 
varies  with  the  absolute  temperature,  the  volume   remaining 
constant. 

43-      VARIATION    OF    PRESSURE,    VOLUME,    AND 
TEMPERATURE  OF  A   SATURATED  VAPOR. 

References. — Edser,  p.  220;  Watson,  p.  251. 

The  apparatus  is  the  same  as  that  used  in  the  experiment 
on  Boyle's  Law.  Instead  of  a  gas,  there  is  in  one  closed  tube 
some  liquid  ether  and  in  the  other  some  water.  (In  some  of 
the  instruments  the  tubes  contain  carbon  bisulphide  and  water 
in  place  of  ether  and  water.  Note  which  you  are  using.)  The 
pressure  on  the  vapor  above  each  liquid  is  produced  by  a  mer- 
cury column  in  a  rubber  U-tube,  one  side  of  which  is  connected 
to  the  closed  tubes,  the  other  being  connected  to  an  open  glass 
tube.  By  measuring  the  height  of  the  mercury  menisci  on  the 
two  sides,  and  noting  the  barometric  pressure,  the  pressure 
on  the  vapor  may  be  obtained.  In  doing  this  there  should  be 
added  to  the  reading  of  the  meniscus  in  the  closed  tube  the 
mercury-equivalent  of  the  height  of  the  liquid  column  in  the 
tube.  The  volume  may  be  read  directly  from  the  graduated 
tube,  and  may  be  changed  by  raising  or  lowering  the  mercury. 
The  temperature  may.  be  regulated  by  means  of  a  water-bath 
surrounding  the  closed  tube. 

(a)  At  the  temperature  of  the  room,  without  any  water 
around  the  closed  tubes,  make  five  different  readings,  chang- 
ing the  volumes  and  reading  the  corresponding  pressures. 


76  VARIATION    01?    A    SATTRATED    VAPOR.  [43 

Record  the  volume  and  pressure  each  time  for  both  closed 
tubes.  After  changing  the  volume  each  time,  wait  until  con- 
ditions become  steady  before  taking  the  readings. 

(b)  Surround  the  tube  with  a  mixture  of  ice  and  water. 
Arranging  the  tubes  so  that  you  can  read  the  mercury  men- 
iscus, repeat  (a),  making,  however,  only  two  readings  of  vol- 
umes and  corresponding  pressures.  Melt  the  ice  gradually 
and  increase  the  temperature  to  5°,  and  take  two  more  read- 
ings, having  one  volume  the  same  as  with  the  temperature  at 
o°.  Keeping  one  volume  the  same  at  each  temperature,  re- 
peat the  readings,  making  two  settings  at  each  temperature, 
with  the  bath  at  TO°,  20°,  30°,  35°,  40°,  and  45°.  Do  not  go 
higher  with  the  instruments  containing  ether.  If  carbon 
bisulphide  be  used,  omit  the  readings  at  35°  but  make  a  read- 
ing at  50°  and  at  55°,  but  at  no  higher  temperature. 

(r)  Put  a  little  ether  (or  carbon  bisulphide,  if  this  be  the 
liquid  in  the  tube)  in  a  test-tube,  and  boil  it  over  a  water-bath 
with  a  thermometer  placed  in  the  ascending  vapor.  Xote  the 
boiling  point.  Obtain  the  boiling  point  of  water  from  the 
Tables. 

(d)  What  relation  is  found  to  exist  in  each  case  between 
the  volume  and  pressure  of  a  saturated  vapor  at  constant  tem- 
perature? Does  Boyle's  Law  hold? 

From  the  observations  for  each  vapor,  plot  on  coordinate 
paper  a  curve,  with  temperatures  as  abscissae  and  correspond- 
ing pressures  as  ordinates,  for  some  volume  which  is  the  same 
in  the  different  sets  of  readings.  How  do  you  find  that  the 
''vapor  pressure"  varies  with  the  temperature,  the  volume  being 
constant  ? 

From  the  curves  of  ether  and  water,  or  carbon  bisulphide 
and  water,  find  the  temperatures  corresponding  to  atmos- 
pheric pressure.  Compare  them  with  the  boiling  points  of 
ether  and  water  found  in  (c).  To  what  is  the  vapor  pressure 
of  any  liquid  equal  at  its  boiling  point? 


44]  HYGROMETRY.  77 

44.     HYGROMETRY. 

References. — Miller,    p.    197;    Millikan,    p.    164;    Watson's    Practical 
Physics,   p.   247;    Edser,  p.   240. 

In  this  experiment  the  dew-point  and  the  relative  and  abso- 
lute humidity  of  the  air  are  to  be  determined.  The  absolute 
humidity,  d,  is  the  density  of  the  water-vapor  present  in  the 
air,  usually  expressed  in  grams  per  cubic  meter.  The  rela- 
tive humidity  is  the  ratio  of  the  amount  of  water  actually  pres- 
ent in  the  air  to  the  amount  required  to  saturate  it  at  the  same 
temperature,  the  latter  quantity  being  the  maximum  amount 
of  water-vapor  that  can  be  held  in  suspension  at  that  tempera- 
ture. The  dew-point  is  the  temperature  at  which  the  amount 
of  water  actually  present  in  the  air  would  saturate  it,  that  is, 
the  temperature  to  which  the  air  must  be  lowered  before  the 
condensation  of  water  will  begin.  The  pressure  of  water- 
vapor  is  the  pressure  which  it  would  exert  by  itself  if  there 
were  no  air  present  in  the  space  considered.  By  Dalton's  Law 
this  is  the  pressure  it  actually  does  exert  when  mixed  with 
air.  In  a  given  volume  the  mass  of  vapor  is  proportional  to  the 
pressure,  so  that  the  relative  humidity  is  equal  to  the  ratio  of 
the  pressure,  p,  of  the  water-vapor  in  the  air  to  the  pressure, 
P,  of  saturated  water-vapor  at  that  temperature,  that  is,  rela- 
tive humidity  is  equal  to  p/P. 

(I)     Regnault's  Hygrometer. 

(a)  Partially  fill  one  of  the  hygrometer  tubes  with  ether 
and  insert  a  thermometer  with  its  bulb  in  the  liquid.  Force  a 
current  of  air  through  the  ether  with  a  bicycle  pump.  The 
rapid  evaporation  of  the  liquid  causes  the  temperature  to  fall. 
When  the  tube  and  the  air  immediately  about  it  are  cooled  to 
the  dew-point,  moisture  appears  on  the  tube,  this  being  detected 
more  easily  by  comparison  with  the  other  tube.  Note  the  tem- 
perature at  which  dew  begins  to  form.  Allow  the  tube  to 
become  warm  and  record  the  temperature  at  which  the  dew 


/8  DENSITY  OF  THE  AIR  BY  THE  BARODEIK.  [45 

disappears.  Take  the  mean  of  these  two  as  the  dew-point. 
Make  three  such  determinations  of  the  dew-point. 

(b)  From  Whiting's  Tables  find  the  pressure  of  saturated 
water- vapor  at  the  dew-point  and  also  at  the  temperature  of 
the  room,  and  calculate  the  relative  humidity.  The  absolute 
humidity  may  be  found  by  multiplying  the  relative  humidity 
by  the  number  of  grams  of  saturated  water-vapor  in  a  cubic 
meter  of  air  at  the  room  temperature,  found  in  Whiting's 
Tables. 

II.  Wet-  and  Dry-Bulb  Hygrometer,  or  Augusta's  Psy- 
chrometer. 

In  the  wet-  and  dry-bulb  hygrometer,  one  bulb  is  covered 
with  wicking  which  dips  into  water,  so  that  the  bulb  is  cooled 
by  evaporation.  After  the  two  thermometers  come  to  constant 
temperatures,  record  the  temperature  of  the  dry  bulb,  t,  and 
of  the  wet  bulb,  t18  Read  the  barometer.  The  following 
empirical  formula  may  then  be  used: 

p  =  pi  —  0.0008  b  ( t  —  tx ) , 

where  p  is  the  pressure  of  water- vapor  present  in  the  atmos- 
phere, pi  the  pressure  of  saturated  vapor  at  the  temperature  of 
the  wet  bulb  (obtained  from  Whiting's  Tables),  and  b  is  the 
barometric  pressure,  all  three  being  expressed  in  millimeters 
of  mercury.  Find  the  pressure  of  saturated  water-vapor  at 
the  room  temperature  from  the  Tables,  and  calculate  the  rela- 
tive humidity.  Find  then  the  absolute  humidity  as  in  (b). 
From  Table  15,  p.  868,  Whiting's  Tables,  find  the  dew-point 
from  the  readings  of  the  wet-  and  dry-bulb  hygrometer. 

Compare  the  values  obtained  in  I  and  II  for  the  humidity 
and  the  dew-point. 

45.     DENSITY  OF  THE  AIR  BY  THE  BARODEIK. 

I.  To  find  the  difference  between  the  barodeik  reading  and 
the  true  density  of  the  air. 

The  barodeik  is  an  ordinary  balance,  having  a  hermetically 


45]  DENSITY  OF  THE  MR  BY  THE  BARODElK.  .          79 

sealed  flask  suspended  from  one  scale-pan,  and  from  the  other 
(as  a  counterpoise)  a  glass  plate  so  chosen  as  to  have  a  surface 
about  equal  to  the  exterior  surface  of  the  flask.  The  reading 
of  the  balance-pointer  on  a  properly  graduated  scale  gives 
the  density  of  the  surrounding  air. 

(a)  Set  and  read  the  barometer  with  great  care.  Read  the 
wet-  and  dry-bulb  hygrometer.  From  Table  15,  Whiting's 
Tables,  calculate  the  dew-point  and  also  the  pressure  of  water- 
vapor  in  the  air.  Remember  that  "dew-point"  means  the 
temperature  at  which  the  water-vapor  now  in  the  air  would  be 
saturated  ;  or  the  temperature  at  which  the  existing  pressure  of 
water-vapor  in  the  air  would  be  the  maximum  pressure. 

(&)  From  (a)  calculate  the  density  of  the  air.  The  mass 
of  one  cu.  cm.  of  dry  air,  at  o°C.  and  76  cm.  pressure,  is 
0.001293  grams.  The  mass  of  the  same  volume  of  water-vapor, 
under  the  same  conditions,  is  5/8  as  much.  Then,  if  H  be 
the  barometric  height,  f  the  pressure  of  water-vapor,  and  t  the 
temperature,  the  mass  of  dry  air  in  one  cu.  cm.  of  moist  air 
is,  by  the  general  gas  law,  PV  =  RmT, 

i        H  —  f 


where  a  is  the  coefficient  of  expansion  of  a  gas.    The  mass  of 
water-vapor  in  the  same  volume  is 

M,  =  (s/8)  0.001293  ^~  ~- 

The  sum  of  these  two  is  the  required  density.      (Deschanel, 
p.  400.) 

(c)  Read  the  barodeik.     Do  not  touch  the  instrument,  but, 
by  moving  the  hand  near  the  flask,  set  up  a  small  vibration  ; 
then  close  the  case,   and   determine  the  resting-point  of  the 
pointer,  which  is  the  density  of  the  air  as  indicated  by  the 
instrument. 

(d)  Record   the   difference   between   the    reading   thus   ob- 


8O  DENSITY  OF  THE  AIR  BY  THE  BARODEIK.  [45 

tained   and  the  true  density   found   in    (b),   with   the   proper 
sign,  so  that  when  added  algebraically  to  the  observed  read- 
ing it  will  give  the  true  density  of  the  air.     This  is  the  abso- 
lute correction  for  the  scale-division  to  which  it  applies. 
II.     Relative  Calibration  of  the  Barodeik  Scale. 

(a)  Read  the  instrument   as   in   I    (c).     Repeat  with  the 
rider  at  division  2  on  its  scale  to  the  right  of  the  center,  which 
is  equivalent  to  adding  2  mg.  to  the  right-hand  pan  of  the 
balance;  then  use  the  rider  in  the  corresponding  position  on 
the  left-hand   side. 

(b)  Repeat  the  readings  with  the  rider  at  division  5 ;  first 
on  the  right-hand,  then  on  the  left-hand  side. 

(c)  Using  the  exterior  volumes  of  flask  and  plate  as  given 
on   the   instrument,   calculate   the   changes   in   the   density   of 
the  air  which  would  produce  the  same  effects  on  the  instru- 
ment as  the  putting  of  the  separate  masses  on  the  right  pan, 
and  on  the  left  pan.     From  these  results  construct  a  table  of 
corrections,  with  the  proper  signs,  for  the  different  resting- 
points  observed.     Note  that  this  is  a  relative  calibration ;  that 
is,  it  gives  the  corrections  to  be  applied  to  certain  readings,  as 
compared  with  one  reading   (namely,  that  when  no  weights 
were  used),  which  is  assumed  correct. 

(rf)  In  part  I  the  absolute  correction  for  a  certain  reading 
was  found.  That  reading  was  the  same  as,  or  not  far  from,  the 
one  assumed  correct  above,  so  the  same  absolute  correction 
may  be  applied  to  the  latter.  By  means  of  this,  convert  the 
table  of  relative  corrections,  (c),  into  a  table  of  absolute 
corrections.  This  completes  the  absolute  calibration  of  the 
instrument. 

(e)  Plot  on  coordinate  paper  the  readings  of  the  barodeik 
scale  as  abscissae  and  the  relative  corrections  of  (c)  as  ordi- 
nates,  but  on  a  much  larger  scale.  Show  how  the  curve  can 
be  made  to  indicate  absolute  corrections  instead  of  relative, 
by  moving  the  horizontal  axis  of  reference  up  or  down  by 


46]  COEFFICIENT  OF  FRICTION.  8 1 

a  proper  amount.  This  converts  it  into  an  absolute  calibra- 
tion curve  for  the  instrument,  enabling  one  to  find  the  density 
of  the  air  at  any  time  by  merely  reading-  the  resting-point  of 
the  pointer. 

46.     COEFFICIENT  OF  FRICTION. 

References. — Watson,   p.    in;   Ferry  and  Jones,  p.   75. 

When  one  body  is  caused  to  slide  over  the  surface  of 
another  the  force  which  is  brought  into  play  to  oppose  the 
motion  is  called  "friction."'  This  force  is  parallel  to  the  sur- 
face and  opposite  in  direction  to  the  motion.  When  the 
sliding  body  is  on  a  level  plane,  the  normal  force  is  equal  to 
the  weight  of  the  body ;  when  on  an  inclined  plane  it  is 
equal  to  the  component  of  the  body's  weight  normal  to  the 
plane.  In  either  case  the  force  of  friction  is  equal  and  oppo- 
site to  the  force  necessary  just  to  produce  motion  (starting 
friction),  or  to  keep  the  body  moving  at  constant  speed 
(moving  friction).  If  P  is  the  normal  force  between  the  two 
surfaces  and  F  is  the  force  of  friction,  the  ratio  F/P  is 
called  the  coefficient  of  starting  or  moving  friction,  as  the 
case  may  be,  and  is  usually  denoted  by  the  Greek  letter  p. 
Ry  measuring  these  forces  and  calculating  their  ratio  the 
coefficient  may  be  determined.  A  second  method  of  deter- 
mining the  coefficient  of  friction  is  to  vary  the  inclination  of 
the  plane  until  the  body  by  its  weight  just  begins  to  move 
(starting  friction)  or  moves  down  the  plane  with  constant 
speed  (moving  friction).  If  the  angle  of  inclination  at 
which  this  occurs  is  a,  prove  that  the  coefficient  of  friction  is 
equal  to  tan  a. 

(a)  The  coefficient  of  friction  is  to  be  found  between 
blocks  provided  and  the  surface  of  a  plane  whose  inclination 
can  be  varied.  Take  one  of  the  blocks  and  weigh  it.  Deter- 
mine the  force  of  starting  friction  and  also  of  moving  friction 


82  COEFFICIENT  OF  FRICTION.  [46 

on  a  level  surface  by  applying  forces  to  it  by  means  of  the 
shot-bucket  and  string  and  pulley.  Calculate  the  coefficient 
of  friction  for  the  two  cases. 

(b)  Determine    the    coefficient    of    friction      for   the    same 
block  and  surface  from  the  tangent  of  the  angle  obtained  by 
varying  the  inclination  of  the  plane  until    ( i )    motion   com- 
mences, and   (2)   continues  at  constant  speed. 

(c)  Set  the  plane  at  the  angle  giving  constant  speed  down 
the  plane,  and  find  the  force  that  will  cause  the  block  to  move 
up  the  plane  at  constant  speed.     Calculate  the  coefficient  of 
friction. 

(d)  Set  the  plane  at  an  angle  of  30°   and  find  the   force 
necessary  to  move  the  block  up  the  plane  at  constant  speed, 
and   then,   if  possible,  the   force  necessary  to  make   it  move 
down  the  plane  at  constant  speed.  Then,  by  calculating  the 
force  perpendicular  to  the  plane,  find  the  coefficient  of  fric- 
tion.   If  this  process  is  not  entirely  clear,  repeat  with  the  plane 
at  an  angle  of  60°. 

(c)  Repeat  (a),  for  starting  friction,  having  the  block 
"loaded"  by  placing  a  known  mass  on  top  of  it.  Compare 
the  coefficient  of  friction  found  with  that  found  in  (a). 

(/)  Take  a  block  having  three  or  more  surfaces  of  differ- 
ent areas  but  of  the  same  smoothness,  and  determine  (by  any 
method)  the  force  of  friction  as  the  block  slides  or  is  moved 
successively  on  the  three  surfaces. 

(g)  Take  a  block  with  surfaces  of  different  degrees  of 
smoothness,  and  determine  the  coefficient  of  starting  friction 
for  two  or  more  sides. 

(h)  Compare  the  results  obtained  from  (a),  (b),  (c), 
(rf),  and  (c),  stating  your  conclusions.  What  do  you  con- 
clude from  (/)  ?  From  (g)  ?  Upon  what  does  the  friction 
between  two  surfaces  depend  ? 


47]  CONSERVATION   OF   MOMENTUM.  83 

47.     CONSERVATION    OF    MOMENTUM.      COEFFI- 
FICIENT  OF  RESTITUTION. 

References.  —  Ames  and  Bliss,  p.  90;  Millikan,  p.  58. 

In  any  system  of  bodies  not  acted  upon  by  any  outside 
force,  and  in  which  the  several  bodies  may  be  moving  with 
different  velocities  and  in  different  directions  with  frequent 
collisions,  the  vector  sum  of  the  momenta  remains  constant. 
This  is  known  as  the  Law  of  Conservation  of  Momen- 
tum. In  our  present  study  the  number  of  bodies  will  be 
limited  to  two  and  the  velocities  restricted  to  the  same  straight 
line,  the  collisions  taking  place  centrally.  Let  us  suppose 
that  we  have  two  bodies,  A  and  B,  suspended  by  strings  so 
that  they  hang  in  contact  when  at  rest.  Let  A  be  dfiawn 
aside  and  then  released.  At  the  lowest  point  of  its  swing  it 
strikes  the  ball  B.  Let  mt  be  the  mass  of  A  and  i^  its  veloc- 
ity just  as  it  strikes  B.  Its  momentum  then  at  this  instant 
is  mlul.  The  ball  B  will  instantly  start  off  with  a  velocity 
v2,  say,  and  a  momentum  m2v2,  if  m2  is  its  mass.  The  ball 
A  may  continue  on  with  a  diminished  velocity,  vx  ;  or  remain 
at  rest,  if  it  loses  all  of  its  momentum  ;  or  it  may  rebound, 
in  which  case  v,  is  negative.  After  impact  the  two  balls  will 
move  away  from  each  other  with  a  relative  velocity  which 
is  greater  the  greater  their  elasticity.  The  elasticity  is  taken 
into  account  in  a  factor  called  the  "coefficient  of  restitution." 
The  coefficient  of  restitution  is  numerically  equal  to  the  ratio 
of  the  relative  velocities  with  which  the  bodies  move  apart 
after  impact  to  that  with  which  they  approached  each  other 
before  impact,  i.  e.,  it  is  given  by  the  equation, 


CD         e  =      = 

"i  —  u, 

where  the  velocities  before  impact,  u^  and  u2,  and  the  veloc- 
ities after  impact,  vx  and  v2,  are  all  in  the  same  straight  line. 
One  or  more  of  the  velocities  mav  be  negative,  or  the 


84  CONSERVATION  OF  MOMENTUM.  [4- 

particular  value  of  a  velocity  may  be  zero,  as  in  the  case 
just  outlined  lu  =  o.  The  value  of  e  always  lies  between 
zero  and  unity.  For  "perfectly  elastic"  bodies  e=  i.  but  for 
all  actual  bodies  e  <  i.  For  inelastic  bodies  e  =  o.  Tn  any 
case,  whether  the  bodies  are  elastic  or  inelastic,  the  conserva- 
tion of  momentum  holds,  i.  e., 

(2)  mjUj  +  rn2u2  =  m,Vj  +  m2v2. 

This  may  be  verified  by  determining  the  masses  and  the  veloc- 
ities. 

In  order  to  find  the  velocity  of  a  suspended  ball  as  it  col- 
lides, or  just  after  collision,  we  make  use  of  the  fact  that  the 
velocity  is  the  same  as  the  ball  would  have  acquired  if  it  had 
fallen  the  same  vertical  distance  that  it  has  descended  in  its 
swing  before  collision,  or  that  it  has  risen  in  its  swing  after 
collision,  as  the  case  may  be.  Let  the  height  be  h;  then,  as 
the  case  may  be,  u  or  v  equals  V2gh-  If  the  angle  of  the 
half  swing  is  a  and  the  length  of  the  pendulum  is  1,  we  have, 

(3)  u,   or  v,  =  V2gl(I— cos  «)• 

(a)  The  numbers  in  the  scale  of  the  frame  from  which 
the  balls  are  suspended  represent  degrees  of  arc.  First  use 
the  two  large  ivory  balls  and  see  that  they  are  adjusted  so  as 
to  hang  fully  in  contact  and  so  that  their  centers  are  in  line. 
Record  the  zero-reading  for  each  ball.  Draw  one  aside 
through  about  TO°  or  15°  and  fix  it  in  position  with  a  thread. 
Record  the  reading.  Release  it  by  burning  the  thread.  Note 
carefully  the  extremity  of  the  swing  of  each  ball  after  impact. 
This  can  be  done  by  placing  a  slider  in  the  position  for  each 
ball.  Several  trials  will  be  necessary  to  accurately  determine 
these  points.  From  these,  and  the  zero-readings,  the  arcs  of 
the  swings  are  found,  and  then  by  measuring  the  length  of 
the  cord  (to  the  center  of  the  ball)  the  velocities  Uj,  vv  and 
v2  can  be  determined.  Repeat  for  two  other  starting  points 
from  which  the  ball  is  released.  Determine  the  masses  of  the 


48]  YOUNG'S  MODULUS  BY  STRETCHING.  85 

balls,   and  then  calculate   for  the  three  cases  the  momentum 
of  the  system  before  and  after  impact. 

(b)  Use  one  large  ivory  ball  and  one  small  one,  and  repeat 
for  one  or  two  starting  points,  releasing  the  large  ball. 

(c)  Repeat    (b),    reversing   the   process   by    releasing   the 
small  ball. 

(d)  Use  two  lead  balls  and  repeat  for  one  or  two  starting 
points,  or  place  a  layer  of  paraffine  on  each  large  ivory  ball  on 
adjacent  sides  and  use  them  as  inelastic  balls. 

(e)  Calculate  the  coefficient  of  restitution  for  all  the  cases 
above.     Does  it  appear  to  depend  on  the  size  of  the  balls  or 
only  on  the  material? 

For  the  same  cases  calculate  the  percentage  difference  in 
the  momentum  before  and  after  impact.  Does  the  "Conserva- 
tion of  Momentum"  appear  to  hold  equally  well  in  all  cases  ? 

Calculate  the  percentage  loss  of  kinetic  energy  for  each 
case.  What  becomes  of  the  energy  apparently  lost?  Is  the 
loss  greater  in  the  more  or  in  the  less  elastic  bodies? 

48.     YOUNG'S   MODULUS  BY  STRETCHING. 

References. — Watson's    Practical    Physics,    p.    99;    Millikan,    p.    65; 
Ferry  and  Jones,  p.    120. 

Hooke's  Law  states  that  in  elastic  bodies,  within  their  elas- 
tic limits,  the  strain  or  deformation  produced  is  propor- 
tional to  the  stress  or  distorting  force.  In  particular  it  states 
that  if  different  forces  be  applied  to  a  wire,  e.  g.,  by  sus- 
pending it  and  hanging  masses  from  it,  the  amount  of 
stretching  will  be  (within  certain  limits)  proportional  to  the 
applied  force.  For  a  wire  of  any  given  material  the  ratio  of 
the  stress  per  unit  area  of  cross-section  to  the  increase  in  length 
per  unit  length  is  a  constant,  and  is  known  as  Young's  mod- 
ulus. (The  word  stress  strictly  applies  to  the  force  inside 
the  wire  which  opposes  the  distorting  force.) 


86  YOUNG'S  MODULUS  BY  STRETCHING.  [48 

A  wire  is  held  vertically  between  two  clamps.  To  the 
lower  clamp,  C,  is  attached  the  end  of  a  rod  whose  upper 
end  is  loosely  held  in  a  support.  From  the  lower  end  of  the 
wire  or  the  clamp  C,  masses  may  be  suspended  and  the  wire 
stretched.  Above  the  upper  end  of  the  rod  is  a  screw  with 
a  divided  head  so  that  small  fractions  of  a  turn  may  be  read. 
When  the  screw  comes  in  contact  writh  the  rod  an  electric 
bell  rings.  If  the  wire  is  then  stretched,  the  screw  must  be 
advanced  again  before  ringing  will  occur.  In  this  way  the 
change  in  length  is  readily  determined,  if  the  pitch  of  the 
screw  is  known. 

(a)  Hang  first  a  1500  gm.  mass  from  the  wire  and  leave 
it  there   for  all  of  the   zero- readings.     This   will   insure  the 
wire  being  straight  at  the  beginning.     Make  a  setting  with 
the  screw  and  read  it  to  o.oi  mm.  or  less.    Then  increase  the 
load  by  adding  500  gms.  at  a  time   (each  time  reading  the 
screw),  until  the  wire  carries  a  load  of  about  4000  gms. 

(b)  Take  the  masses  off,  500  gms.  at  a  time,  reading  the 
screw  each  time. 

(c)  Repeat   (a)  and   (b)  at  least  once.     Record  the  obser- 
vations in  tabulated  form.     Calculate  and  record  the  elonga- 
tion produced  by  each  500  gms. 

(d)  Measure  the  length  of  the  wire  between  the  clamps, 
and  also  the  diameter  of  the  wire. 

(e)  Repeat    (a),    (&),    (c),    (d)    with   a  wire  of  different 
diameter. 

(f)  From  the  mean  elongation  for  a  stretching  force  of  500 
grams-weight,  calculate  separately  Young's  modulus   for  the 
two  wires.    Does  Hooke's  Law  hold  for  these  wires  according 
to  the  results  in  your  tabular  form?  If  there  is  any  variation 
from  the  law,  assign  a  reason  if  you  can. 


49]  HOOKK'S  LAW  FOR  TWISTING.  87 

49.     HOOKE'S  LAW  FOR  TWISTING.    COEFFICIENT 
OF  RIGIDITY. 

References.  —  Millikan,    p.    71;    Ames    and    Bliss,    p.    168. 

If  a  cylindrical  wire  be  fixed  at  one  end,  and  the  free  end 
be  twisted  about  the  axis  of  the  wire,  no  change  of  volume 
will  occur,  but  the  strain  in  the  wire  is  found  to  be  one  of 
shape  only.  For  a  wire  of  given  material,  length,  and  diam- 
eter, the  force-moment  producing  the  twisting  is  found  to  be 
(within  certain  limits)  proportional  to  the  angle  of  twist. 
This  statement  may  be  deduced  mathematically  from 
Hooke's  Law  which  states  that  in  elastic  bodies  (within 
their  elastic  limits)  the  strain  or  deformation  produced  is 
proportional  to  the  stress  or  distorting  force.  The  mathe- 
matical reasoning  establishing  the  relation  between  the  angle 
of  twist,  the  force-moment  producing  the  twisting,  and  the 
material,  length,  and  radius  of  the  wire  is  not  simple,  involv- 
ing integral  calculus.  The  object  of  this  experiment  is  to 
establish  the  relation  experimentally. 

If  M  is  the  moment  of  the  twisting  force,  a  the  angle  of 
twist  in  radians,  1  the  length  of  the  rod,  and  r  its  radius, 
we  ma  write 


a  ==         , 
?rnr4 

In  the  above  equation,  n  is  constant  for  a  given  material  and 
is  called  the  "coefficient  of  rigidity,''  or  sometimes  the  "mod- 
ulus of  torsion." 

(a)  The  torsion  lathe,  as  described  in  the  first  reference 
given,  will  be  used.  The  support  with  graduated  wheel,  and 
also  the  fixed  support,  should  be  clamped  firmly  to  the  table. 
Clamp  the  smallest  rod  provided,  with  one  end  in  the  wheel 
and  the  other  in  the  fixed  support,  being  careful  to  have  the 
rod  straight  and  the  scale  on  the  wheel  adjacent  to  the  ver- 
nier. Add  masses  to  the  pan,  preferably  100  grams  at  a 


88  CENTRIPETAL  FORCK.  [50 

time,  and  read  the  angle  of  twist  for  each  twisting  moment 
used.  Reverse  the  process,  reading  the  angle  each  time  a 
mass  is  taken  off.  Record,  in  tabular  form,  the  masses  used, 
the  corresponding  angular  deflections,  and  the  increase  in 
deflection  for  each  loo-gram  mass  added.  Measure  the  diam- 
eter of  the  wheel,  the  diameter  of  the  rod,  and  the  length  of 
the  rod  between  the  clamps. 

(b)  Repeat  the  observations  of  (o)  with  the  same  rod 
clamped  so  as  to  use  one-half  the  length  there  used. 

(r)  Repeat  with  a  rod  of  the  same  material  and  length  as 
that  used  in  (a),  but  having  a  different  radius. 

Repeat  with  a  rod  of  different  material,  but  having  the 
same  length  and  radius,  approximately,  as  that  used  in  (a). 

(d)  From  your  results  show  how  the  angle  of  twist  varies 
with  the  twisting  moment,  with  the  length  of  the  rod,  and 
with  the  radius  of  the  rod. 

Calculate  the  coefficient  of  rigidity  for  each  case,  (a),  (&), 
(c),  using  the  average  angular  deflection  corresponding  to 
the  force-moment  produced  by  the  weight  of  100  grams. 

If  the  radius  of  the  wire  be  measured  to  the  nearest  o.oi 
mm.,  with  what  accuracy  should  the  length  be  measured? 

50.     CENTRIPETAL  FORCE. 
Reference. — Millikan,  p.  100. 

The  object  of  this  experiment  is  to  determine  the  force 
necessary  to  keep  a  body  of  given  mass  in  a  circle  of  given 
radius,  while  it  moves  with  constant  speed.  Experience 
shows  that  a  body  in  motion  will  continue  to  move  with  the 
same  speed  in  the  same  straight  line,  unless  acted  upon  by  some 
outside  force.  An  outside  force,  if  acting  in  the  direction  of 
the  motion,  will  cause  a  change  in  speed ;  if  acting  at  right 
angles  to  the  direction  of  motion,  it  will  cause  no  change  in 
speed,  but  will  cause  a  change  in  the  direction  of  the  motion.  A 
body  in  motion  always  moves  in  a  straight  line,  unless  there 


50]  CENTRIPETAL   FORCE.  89 

is  a.  force  applied  causing  it  to  leave  the  straight  line.  If 
the  force,  perpendicular  to  the  line  of  the  motion,  be  momentar- 
ily supplied,  the  direction  of  the  motion  is  changed,  but  the 
body  continues  to  move  in  a  straight  line  at  an  angle  with 
its  former  direction.  If  the  force  be  continually  supplied,  the 
body  moves  in  a  curved  path.  If  the  body  be  kept  in  a  circu- 
lar path,  a  force  of  definite  magnitude  must  be  continuously 
applied  to  the  body,  the  direction  of  the  force  being  always  per- 
pendicular to  the  instantaneous  direction  of  the  motion. 
Since  the  instantaneous  direction  of  motion  is  along  the  tan- 
gent, the  force  perpendicular  to  the  direction  of  motion  must 
be  along  the  radius  of  the  circle.  If  the  force  ceases  to  be 
supplied,  the  body  ceases  to  leave  the  straight  line  and  hence 
continues  to  move  in  the  tangent  to  the  circle  at  the  position 
occupied  by  the  body  at  the  instant  the  force  ceased  to  act. 
This  is  illustrated  by  whirling  a  stone  at  the.  end  of  a  string — 
the  string  supplies  the  force  necessary  to  keep  the  stone  in 
a  circular  path.  If  the  string  breaks,  the  necessary  force  is 
no  longer  supplied,  and  the  stone  is  no  longer  pulled  out  of 
the  straight- line  path.  It  moves  away,  therefore,  at  a  tangent 
to  its  former  circular  path.  This  cental  force  is  called  the 
Centripetal  Force,  or  the  Normal  Force.  It  is  called  the 
normal  force  because  it  is  always  normal  to  the  curved  path. 
It  is  always  directed  toward  the  concave  side  of  the  curve. 
If  the  path  is  a  circle,  it  is  directed  inward  along  the  radius. 

For  a  circular  motion  the  magnitude  of  the  normal  force 
is  directly  proportional  to  the  mass  of  the  body  and  the 
square  of  its  speed,  and  inversely  proportional  to  the  radius 
of  the  circle.  This  arises  from  the  fact  that  the  force  is  pro- 
portional to  the  product  of  the  mass  and  the  acceleration  of 
the  body,  and  the  acceleration  is  inward  along  the  radius  and 
of  magnitude  v2/r,  where  v  is  the  speed  of  the  body  and  r  is 
the  radius  of  the  circle. 

(a)  To  a  rotator  is  attached  the  "centripetal  force"  appar- 
atus. Two  masses,  ml  and  m2,  are  arranged  to  slide  along 


9O  FRICTION    BRAKE.       POWER    SUl'lM.lKD    I'.V    A    MOTOR.  [51 

the  horizontal  guides.  They  are  attached,  by  means  of  cords 
passing  over  pulleys,  to  a  large  mass,  M,  which  can  slide 
up  and  down  along  the  vertical  rod.  As  the  speed  of  rota- 
tion is  increased,  more  and  more  force  must  be  supplied  to 
irij  and  m2  in  order  to  hold  them  to  a  circular  path.  Finally, 
when  the  speed  passes  a  certain  value,  the  force  necessary  to 
keep  the  masses  moving  in  their  circular  paths  is  greater  than 
the  weight  of  M  can  supply,  and  the  mass  M  is  lifted. 

The  speed  may  be  so  regulated  that  M  remains  about  half- 
way up  the  rod.  Its  weight,  Mg  dynes,  represents  the  nor- 
mal or  centripetal  force  supplied  to  the  masses  mx  and  m.2. 
Write  the  equation  representing  this  relation.  The  masses 
M,  mlt  and  m2  must  be  determined,  and  the  distances  of  ml 
and  m2  from  the  axis  of  rotation.  The  speeds  of  m,  and  m2 
may  be  calculated,  provided  the  number  of  rotations  in  a 
given  time  be  counted.  Make  several  determinations  of  the 
speed,  varying  its  value  by  altering  the  masses,  mx  and  m,, 
or  by  changing  their  distances,  i^  and  r2,  from  the  axis. 

(b)  In  each  case  test  the  equality  of  the  weight,  Mg,  and 
the  calculated  centripetal  force  required.  Determine,  in  each 
case,  the  percentage  difference.  Point  out  the  principal 
sources  of  error  in  the  experiment. 

In  the  case  of  circular  motion  what  term  is  commonly 
applied  to  the  reaction  against  the  centripetal  force? 

51.     FRICTION   BRAKE.     POWER   SUPPLIED   BY  A 

MOTOR. 

Reference. — Watson,  p.   116. 

The  object  of  this  experiment  is  to  measure  by  means  of 
a  friction  brake  the  power  delivered  by  an  electric  motor  and 
to  study  the  effect  of  altering  the  friction  of  the  different 
parts.  An  electric  motor,  a  bank  of  incandescent  lamps 
arranged  in  parallel,  and  a  key  are  connected  in  series  with 


51  ]          FRICTION    BRAKE.       POWER    SUPPLIED   BY   A    MOTOR.  91 

the  no-volt  power-circuit.  The  circuit  is  made  by  pressing 
the  key.  The  resistance  can  be  decreased  by  introducing  more 
lamps  into  the  circuit.  A  Prony  brake  is  used.  The  Prony 
brake  consists  of  a  lever,  one  end  of  which  is  bound  around 
a  revolving  shaft  in  such  a  way  that  the  friction  produced 
will  tend  to  rotate  the  lever  in  the  direction  in  which  the 
shaft  revolves.  This  tendency  to  rotate  is  balanced  by  a 
spring  balance  acting  at  right  angles  to  the  lever  or  by  the 
weight  of  masses  hung  from  the  lever.  If  P  is  the  force  in 
dynes  acting  on  the  lever  to  prevent  rotation,  and  L  the 
distance  from  the  line  of  P  to  the  center  of  the  shaft,  the 
power  absorbed  by  the  brake,  or  the  work  per  second,  will 
be  27rLnP,  where  n  is  the  number  of  revolutions  of  the  shaft 
per  second. 

(a)  Suspend   a   spring  balance   from   the   iron   stand,   and 
then  attach  it  below  to  the  lever  of  fhe  brake  so  that,  when 
the  motor  is  running,  the  balance  will  oppose  any  tendency 
of  the  brake  to  rotate.     Note  the  reading  of  the  balance  when 
the  motor  is  not  running.     Then  start  the  motor  by  gradually 
decreasing   the    resistance   given   by   the   incandescent   lamps, 
and,  with  the  motor  running  at  less  than  full  speed,  tighten 
the  belt  connecting  the  motor  to  the  shaft     of     the     brake. 
Allow  the  motor  to  run  at  full  speed  with  the  belt  taut,  and 
record  the  number  of  revolutions  of  the  shaft  in  three  min- 
utes, as  given  by  the  speed  counter.     Note  the  reading  of  the 
spring  balance  while  the  shaft  is  rotating.     Take  two  more 
readings  with  the  balance  at  different  points  along  the  lever. 

(b)  Tighten  the  screws  which  bind  the  wooden  blocks  of 
the  brake  against  the  shaft,  and  take  measurements  with  three 
different  lever  arms.     Note  if  the  lamps  grow  brighter  when 
the  friction  is  increased.     If  so,  what  can  be  said  about  the 
dependence  of  the  power  consumed  by  a  motor  on  the  load? 
The  effect  may  also  be  observed  by  tightening  the  belt  con- 
necting motor  and  brake-shaft. 

(c)  Calculate  the  power  delivered  by  the  motor  for  each 


92  -      ABSORPTION    AND   RADIATION.  [52 

of  the  six  measurements.  In  what  units  is  the  power  ex- 
pressed, if  the  force  of  the  balances  is  in  dynes  and  the  lever 
arm  in  centimeters?  Reduce  the  results  to  horse-power.  If 
you  know  the  method  by  which  electrical  power  is  computed, 
show  how  the  efficiency  of  the  motor  may  be  calculated. 

(d)  Disconnect  the   friction   brake,   attach  the   spring  bal- 
ances to  a  cord,  and  hold  or  suspend  them  above  the  motor 
so  that  they  will  pull  in  parallel  lines,  thereby  pressing  the 
cord  against  half  of  the  periphery  of  the  motor-wheel.     Allow- 
ing the  motor  to  run  at  moderate  speed,  record  the  difference 
in  the  readings  of  the  two  spring  balances  as  the  cord  presses 
against  the  wheel. 

(e)  Repeat   (d)    for  the  other  pulley-wheel  on  the  motor- 
shaft,  exerting  as  nearly  as  possible  the  same  tension  as  be- 
fore.    Measure  the  diameter  of  each  of  the  wheels  and  see 
what   relation  exists  between  the  friction  and  the  radius  of 
the  wheels,  the  angle  of  contact  being  the  same  in  the  two 
cases. 

52.     ABSORPTION  AND  RADIATION. 
References. — Watson,  p.  304;   Edser,  p.  436. 

It  is  well  known  that  a  dull  black  surface  absorbs  light 
more  readily  than  a  white  or  light-colored  one.  This  is 
shown  by  the  difficulty  in  illuminating  a  photographic  dark 
room  or  a  room  with  dark-colored  hangings.  The 
purpose  of  this  experiment  is  to  see  whether  the  relations 
which  hold  for  light  apply  also  to  the  vibrations  of  longer 
period  which  are  manifest  to  our  senses  only  through  the 
sensation  of  heat.  That  is,  it  is  proposed  to  study  the  rate 
of  absorption  of  heat  by  black  and  by  polished  surfaces,  and 
also  the  rate  at  which  heat  is  radiated  by  these  surfaces  to  a 
colder  body. 

(a)  A  box  lined  with  tin  has  an  opening  in  the  side  in 
which  three  thermometers  may  be  set  and  read  from  the 


53]  RATIO  OF  THE:  SPECIFIC  HEATS  OP  A  GAS.  93 

outside.  The  bulb  of  one  of  the  thermometers  is  bare,  another 
is  silvered,  and  the  third  is  coated  with  lampblack.  All  three 
thermometers  should  register  the  temperature  of  the  room. 
Record  the  room  temperature.  Heat  water  to  boiling  in  a 
kettle  and  pour  into  the  vessel  in  the  box,  arranging  this  so 
that  the  steam  will  not  reach  the  thermometer  bulbs  and  con- 
dense on  them.  Record  the  readings  of  all  three  thermometers 
each  minute  until  a  steady  temperature  is  reached.  Then  at 
an  even  minute  remove  the  hot  water  and  continue  the  read- 
ings till  the  thermometers  again  register  the  temperature  of 
the  room. 

(b]  Make  a  good  freezing  mixture  in  a  large  beaker,  and 
place   this    in   the   box   close   to  the   thermometer   bulbs,   the 
thermometers   being   equally   distant   from   the   freezing  mix- 
ture.    Read   the  temperatures   each   minute   until   they   cease 
to    fall.      Remove   the    freezing   mixture   and    read   the   ther- 
mometers as  they  return  to  room  temperature. 

(c)  Plot  the  results  of   (a)   and   (b)   on  coordinate  paper, 
using  times  as  abscissae  and  temperatures  as  ordinates,  mak- 
ing the  scale  as  large  as  possible.     Discuss  the  form  of  the 
curves  and  the  relation  between  the  several  curves.     What  re- 
lation exists  between  absorption  and  radiation  at  the  highest 
and   at   the   lowest   temperatures    reached?     Connect   the   re- 
sults with  the  fact  that  stoves  are  made  black  and  the  fender 
and  knobs  of  the  stove  are  nickeled. 


53.     RATIO    OF    THE   TWO    PRINCIPAL    SPECIFIC 
HEATS  OF  A  GAS. 

References. — Watson,   p.   328;    Watson's    Practical   Physics,  p.   267. 

The  object  of  this  experiment  is  to  obtain,  by  the  appli- 
cation of  the  laws  of  thermodynamics,  the  value  of  y,  the 
ratio  of  the  specific  heat  of  a  gas  at  constant  pressure  to  its 
specific  heat  at  constant  volume.  Air  is  compressed,  adia- 


94  KATIO  OF  THK  SI'KCIKIC   HKATS  <>!•'  A  C.AS.  [53 

hatically,  in  a  large  balloon-flask  to  a  pressure  greater 
than  atmospheric  pressure,  and  is  allowed  to  cool  to  room 
temperature,  tr  The  pressure,  p,,  is  then  measured  on  the 
manometer.  Let  the  volume  of  unit  mass  of  gas  in  the  flask- 
be  v,.  The  stop-cock  is  now  opened  and  the  air  allowed  to 
expand,  the  volume,  v,,  increasing  to  v2,  the  pressure  falling 
to  atmospheric  pressure,  p0,  and  the  temperature  to  t2.  The 
stop-cock  is  again  closed  and  the  temperature  allowed  to  rise 
again  to  tlf  the  volume  remaining  the  same,  v2.  and  the  pres- 
sure rising  to  p2.  The  gas  has  now  been  in  three  conditions, 
as  follows : 

Condition.  Pressure.  Vol.  of  i  gm.  Temperature. 

I.  p,  Y!  t: 

II.  Po  V2  t, 

III.  p2  V,  t, 

The  change  from  I  to  II  was  adiabatic  (i.  e.,  no  heat  passed 
in  or  out ;  see  Watson,  p.  326) ,  hence 

(1)  1W  =  p0v./,  or    (V1/v2)>'=p0/p1. 

The  change  from  I  to  III  is  isothermal,  hence,  by  Boyle's  Law, 

(2)  PiV1  =  p2v2,   or    (Vl/v2K=  (Po/Pi)''. 
Hence,   (p2/p,)y=  (Po/Pi)  '<  or,  solving  for  y, 

log  Po- log  P. 
log  p,  —  log  p, 

Perform  the  experiment  as  indicated  above,  making  several 
determinations  of  the  pressures,  reading  the  manometer  and 
the  barometer.  Calculate  y.  Take  the  value  given  in  Watson 
(p.  325)  as  correct,  and  calculate  the  percentage  error  of 
vour  result. 


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